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

Q (n, q)

of aggregate claim amount of n i.i.d. risks by a bilinear function of n and

\sqrt{n}

where the solvency coefficient

k_{q}

has to be determined and Y is the generic individual claim severity variable.

E (\cdot)

and

σ (\cdot)

are a mean and standard deviation of a randon variable. Using the approximation the solvency capital requirement SCR is deduced as

g_{j} (x_{j}) = \frac{K_{j} + n_{j} (x_{j} - π_{j}) (1 - e_{j})}{k_{q} σ (Y) \sqrt{n}} \geq 1,

\begin{matrix} X_{j} : = {x_{j} \in [\underline{x}, \bar{x}] | g_{j} (x_{j}) \geq 0} \\ = {x_{j} \in [\underline{x}, \bar{x}] | K_{j} + n_{j} (x_{j} - π_{j}) (1 - e_{j}) \geq k_{q} σ (Y) \sqrt{n}}, \end{matrix}

and

\underline{x}, \bar{x}

are the minimum and the maximum premium, respectively.

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

z (k) \in ℝ^{q}

be economic factor in kth period and

γ_{i j} (k) \in ℝ^{q}

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 [6] ) describing insureds’ movement to insurers.

P (k) = (\begin{matrix} p_{1, 1} (k) & p_{1, 2} (k) & \dots & p_{1, I} (k) & p_{1, I + 1} (k) \\ p_{2, 1} (k) & p_{2, 2} (k) & \dots & p_{2, I} (k) & p_{2, I + 1} (k) \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ p_{I, 1} (k) & p_{I, 2} (k) & \dots & p_{I, I} (k) & p_{I, I + 1} (k) \\ p_{I + 1, 1} (k) & p_{I + 1, 2} (k) & \dots & p_{I + 1, I} (k) & p_{I + 1, I + 1} (k) \end{matrix}),

where

p_{i, j} (k)

denotes the probability for customers to switch from insurer i to j in kth period.

(I + 1)

th column corresponds to uninsured ones whose state can be called inactive. According to [7] (see, also [1] ), the transition probability can be modelled as

p_{i, j} (k) = (\begin{array}{l} \frac{1}{1 + \sum_{l = 1}^{I} \exp (〈 γ_{i l} (k), z (k) 〉)} & if j = I + 1 \\ \frac{\exp (〈 γ_{i j} (k), z (k) 〉)}{1 + \sum_{l = 1}^{I} \exp (〈 γ_{i l} (k), z (k) 〉)} & if j \neq I + 1, \end{array}

where

〈 \cdot, \cdot 〉

is the Euclidean inner product. If the economy is deteriorate in kth period, some insureds don’t want to keep insurance contracts, therefore

p_{i, j} (k), j = 1, \dots, I

will decrease and

p_{i, I + 1}

will increase. In kth period, the portfolio size

N_{j} (k)

of insurer j for the next period is determined by the sum of renewed policies and businesses coming from other insurers. Hence

\begin{matrix} N_{j} (k) = N_{j} (k - 1) p_{j, j} (k) + \sum_{i = 1, i \neq j}^{I + 1} N_{i} (k - 1) p_{i, j} (k) \\ = \sum_{i = 1}^{I + 1} N_{i} (k - 1) p_{i, j} (k), \end{matrix}

where

N_{j} (0) = n_{j}

. Let

r_{f}

be a risk free rate and

v = \frac{1}{1 + r_{f}}

be a discount

factor. Based on [1] , the insurer j maximizes the present value of expected profit of renewing policies defined as

O_{m}^{j} (x) = \sum_{k = 1}^{m} \frac{v^{k} N_{j} (k)}{n} (1 - β_{j} (k) (\frac{x_{j}^{k}}{m_{j} (x^{k})} - 1)) (x_{j}^{k} - π_{j} (k)),

(1)

where

x = {(x_{1}^{T}, \dots, x_{I}^{T})}^{T}

x_{j} = {(x_{j}^{1}, \dots, x_{j}^{m})}^{T}

for

j = 1, \dots, I

, and

m_{j} (x^{k}) = \frac{1}{I - 1} \sum_{i \neq j} x_{i}^{k}

. The solvency constraints for insurer j can be redefined as

g_{l}^{j} (x_{j}) = \frac{K_{j} + \sum_{k = 1}^{l} v^{k} N_{j} (k) (x_{j}^{k} - π_{j} (k)) (1 - e_{j} (k))}{k_{q} σ (Y) \sqrt{\sum_{k = 1}^{l} v^{k} N_{j} (k)}} - 1, for l = 1, 2, \dots, m .

\begin{array}{l} X_{m}^{j} : = {x_{j} \in {[\underline{x}, \bar{x}]}^{m} | g_{l}^{j} (x_{j}) \geq 0, l = 1, \dots, m} \\ = {x_{j} \in {[\underline{x}, \bar{x}]}^{m} | K_{j} + \sum_{k = 1}^{l} v^{k} N_{j} (k) (x_{j}^{k} - π_{j} (k)) (1 - e_{j} (k)) \\ \geq k_{q} σ (Y) \sqrt{\sum_{k = 1}^{l} v^{k} N_{j} (k)}, l = \bar{1, m}} \end{array}

(2)

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 [1] and by Theorem 1 in [8] , we can verify the existence of a Nash equilibrium. On the other hand, since for any

(x_{j}^{1}, \dots, x_{j}^{l - 1}, x_{j}^{l + 1}, \dots, x_{j}^{m}; x_{- j})

, the function

O_{m}^{j} (x)

is strictly concave and differentiable with respect to

x_{j}^{l}

, for

\forall x, y \in ℝ^{I m}

it hold

〈 \nabla_{x_{j}} O_{m}^{j} (x), y_{j} - x_{j} 〉 > O_{m}^{j} (y) - O_{m}^{j} (x),

〈 \nabla_{x_{j}} O_{m}^{j} (y), x_{j} - y_{j} 〉 > O_{m}^{j} (x) - O_{m}^{j} (y) .

\begin{array}{l} 〈 \nabla_{x_{j}} O_{m}^{j} (x), y_{j} - x_{j} 〉 + 〈 \nabla_{x_{j}} O_{m}^{j} (y), x_{j} - y_{j} 〉 \\ > O_{m}^{j} (y) - O_{m}^{j} (x) + O_{m}^{j} (x) - O_{m}^{j} (y) = 0. \end{array}

Denoting by

r = {(1, \dots,1)}^{T} \in ℝ^{I}

and taking the sum by

j, j = 1, \dots, I

, we obtain that

\sum_{j = 1}^{I} r_{j} 〈 \nabla_{x_{j}} O_{m}^{j} (x), y_{j} - x_{j} 〉 + \sum_{j = 1}^{I} r_{j} 〈 \nabla_{x_{j}} O_{m}^{j} (y), x_{j} - y_{j} 〉 > 0, \forall x, y \in ℝ^{m I}

which guarantees the uniqueness of the equilibrium (cf. Theorem 2 in [8] ).

□

Proposition 2. Let

x^{*}

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

x_{j}^{k *} \in] \underline{x}, \bar{x} [

depends on the parameters in the following way:

a) It increases with break-even premiums

π_{j} (k)

, solvency coefficient

k_{q}

, loss volatility

σ (Y)

, expense rate

e_{j} (k)

, and risk free rate

r_{f}

for

k \geq 2

and

b) Decreases with sensitivity parameter

β_{j} (k)

, capital

K_{j}

for

k = 1

, and, portfolio size

N_{j} (l), l = 1, \dots, k

for

k \geq 2

2) If all constraint functions are inactive, then the premium equilibrium is a solution of the linear system of equations

M_{β} = (\begin{matrix} A_{1} & 0 & 0 & \dots & 0 \\ 0 & A_{2} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & A_{m} \end{matrix}),

v = {(β_{1} (1) π_{1} (1), \dots, β_{I} (1) π_{I} (1), \dots, β_{1} (m) π_{1} (m), \dots, β_{I} (m) π_{I} (m))}^{T},

A_{k} = (\begin{matrix} 2 β_{1} (k) & - \frac{1 + β_{1} (k)}{I - 1} & \dots & - \frac{1 + β_{1} (k)}{I - 1} \\ - \frac{1 + β_{2} (k)}{I - 1} & 2 β_{2} (k) & \dots & - \frac{1 + β_{2} (k)}{I - 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ - \frac{1 + β_{I} (k)}{I - 1} & - \frac{1 + β_{I} (k)}{I - 1} & \dots & 2 β_{I} (k) \end{matrix}), k = 1, \dots, m .

Proof: The KKT conditions for the premium equilibrium

x_{j}^{*}

of insurer j has the following form:

{\begin{array}{l} \nabla_{x_{j}} O_{j} (x^{*}) + \sum_{l = 1}^{m} λ_{1 l}^{j *} \nabla_{x_{j}} g_{l}^{j} (x_{j}^{*}) + λ_{2}^{j *} - λ_{3}^{j *} = 0, \\ λ^{j *} = {(λ_{1}^{j *}, λ_{2}^{j *}, λ_{3}^{j *})}^{T} \geq 0, λ_{1}^{j *} \in ℝ^{m}, λ_{2}^{j *} \in ℝ^{m}, λ_{3}^{j *} \in ℝ^{m}, \\ g_{k}^{j} (x_{j}^{*}) \geq 0, x_{j}^{k *} - \underline{x} \geq 0, \bar{x} - x_{j}^{k} \geq 0, k = 1, \dots, m, \\ λ_{1 k}^{j *} g_{k}^{j} (x_{j}^{*}) = 0, λ_{2 k}^{j *} (x_{j}^{k *} - \underline{x}) = 0, λ_{3 k}^{j *} (\bar{x} - x_{j}^{k}) = 0, k = 1, \dots, m . \end{array}

\frac{\partial}{\partial x_{j}^{k}} O_{m}^{j} (x^{*}) + \sum_{l = 1}^{m} λ_{1 l}^{j *} \frac{\partial}{\partial x_{j}^{k}} g_{l}^{j} (x_{j}^{*}) + λ_{2 k}^{j *} - λ_{3 k}^{j *} = 0.

(3)

1) Let

x_{j}^{k *} \in] \underline{x}, \bar{x} [

. Then

λ_{2 k}^{j *} = λ_{3 k}^{j *} = 0

. We consider two cases.

a) Let us assume that the solvency constraints are all inactive, i.e.,

g_{l}^{j} (x_{j}^{*}) > 0, l = 1, \dots, m

. Then, insurer j’s premium equilibrium verifies

\frac{\partial}{\partial x_{j}^{k}} O_{m}^{j} (x^{*}) = 0

, i.e.,

\frac{v^{k} N_{j} (k)}{n} (1 - 2 β_{j} (k) \frac{x_{j}^{k *}}{m_{j} (x^{k *})} + β_{j} (k) + β_{j} (k) \frac{π_{j} (k)}{m_{j} (x^{k *})}) = 0.

(4)

Let

x_{j}^{k} (y) : = {(x_{j}^{1}, \dots, x_{j}^{k - 1}, y, x_{j}^{k + 1}, \dots, x_{j}^{m}; x_{- j})}^{T}

. In order to investigate the sensitivity depending on parameter z, let us define the function

F_{x}^{k, j}

F_{x}^{k, j} (z, y) : = \frac{\partial}{\partial x_{j}^{k}} O_{m}^{j} (x_{j}^{k} (y), z),

and consider the equation of the form

F_{x}^{k, j} (z, y) = 0

. Under assumptions that partial derivatives of

F_{x}^{k, j}

exist and are continuous at

(z_{0}, y_{0})

, and also

\frac{\partial F_{x}^{k, j}}{\partial y} (z_{0}, y_{0}) \neq 0

, by the implicit function theorem, there exists a function

φ

defined in a neighbourhood of

(z_{0}, y_{0})

such that

F_{x}^{k, j} (z, φ (z)) = 0

and

φ (z_{0}) = y_{0}

. The derivative of

φ

is given by

ϕ^{'} (z) = - {\frac{\frac{\partial F_{x}^{k, j}}{\partial z}}{\frac{\partial F_{x}^{k, j}}{\partial y}} |}_{y = φ (z)} .

\frac{\partial F_{x}^{k, j}}{\partial y} (z, y) = \frac{\partial^{2} O_{m}^{j}}{\partial x_{j}^{k 2}} = - 2 β_{j} (k) \frac{v^{k} N_{j} (k)}{n \cdot m_{j} (x^{k})} < 0.

sign (ϕ^{'} (z)) = sign (\frac{\partial F_{x}^{k, j}}{\partial z} (z, ϕ (z))) .

\frac{\partial F_{x}^{k, j}}{\partial z} (z, y) = \frac{v^{k} N_{j} (k) β_{j} (k)}{n \cdot m_{j} (x^{k})} > 0.

In other words, the function

π_{j} (k) \to x_{j}^{k *} (π_{j} (k))

is increasing.

\frac{\partial F_{x}^{k, j}}{\partial z} (z, y) = \frac{v^{k} N_{j} (k)}{n} (- 2 \frac{y}{m_{j} (x^{k})} + 1 + \frac{π_{j} (k)}{m_{j} (x^{k})}) .

\frac{\partial F_{x}^{k, j}}{\partial z} (z, ϕ (z)) = \frac{v^{k} N_{j} (k)}{n} \cdot \frac{- 1}{z} < 0.

Therefore, the function

β_{j} (k) \to x_{j}^{k *} (β_{j} (k))

is decreasing.

b) If the solvency constraints are all active, then the premium equilibrium satisfies

g_{l}^{j} (x_{j}^{*}) = 0

, for

l = 1, \dots, m

and consequently, one get

x_{j}^{1 *} = π_{j} (1) + \frac{k_{q} σ (Y) \sqrt{v N_{j} (1)} - K_{j}}{v N_{j} (1) (1 - e_{j} (1))},

x_{j}^{l *} = π_{j} (l) + \frac{k_{q} σ (Y)}{(\sqrt{\sum_{k = 1}^{l} v^{k} N_{j} (k)} + \sqrt{\sum_{k = 1}^{l - 1} v^{k} N_{j} (k)}) (1 - e_{j} (l))}, l = 2, \dots, m .

(5)

From (5), we can verify directly that

x_{j}^{k *}

is an increasing function of

π_{j} (k)

k_{q}

e_{j} (k)

and

r_{f}

for

k \geq 2

. Moreover, it is a decreasing function of

K_{j}

for

k = 1

and

N_{j} (l), l = 1, \dots, k

for

k \geq 2

2) If all constraints are inactive at a Nash equilibrium

x^{*}

, then taking into account

m_{j} (x^{k}) = \frac{1}{I - 1} \sum_{i \neq j} x_{i}^{k}

and from (4) follows that

2 β_{j} (k) x_{j}^{k *} - (1 + β_{j} (k)) \frac{1}{I - 1} \sum_{i \neq j} x_{i}^{k *} = β_{j} (k) π_{j} (k), \forall j, k .

This system can be rewritten in matrix form as

M_{β} x = v

. As in [1] mentioned, we can see that the matrix

M_{β}

is strictly diagonally dominant if the conditions

β_{j} (k) > 1, j = \bar{1, I}, k = \bar{1, m}

are fulfilled. Under this condition

M_{β}

is invertible and therefore

x^{*} = M_{β}^{- 1} v

Remark 1. If

x_{j}^{k *} = \underline{x}

\bar{x}

, then the premium equilibrium is independent of those parameters.

Remark 2. For a game with one leader and

I - 1

followers with payoff functions

O_{m}^{j}

and the strategy set

X_{m}^{j}

, a Stackelberg equilibrium is the problem that consists in finding a vector

\bar{x} = ({\bar{x}}_{1}^{T}, \dots, {\bar{x}}_{I}^{T})

{\bar{x}}_{j} = ({\bar{x}}_{j}^{1}, \dots, {\bar{x}}_{j}^{m})

such that

{\bar{x}}_{1}

solves the problem

where

(x_{2}, \dots, x_{I})

is a Nash equilibrium for the game with the

I - 1

followers and given strategy

x_{1}

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. [1] ).

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

x \in Ω : = X_{m}^{1} \times X_{m}^{2} \times \dots \times X_{m}^{I}

such that

where

F (x) = {(\nabla_{x_{j}} O_{m}^{j} (x))}_{j = 1}^{I}

. In order solve the problem (VI), we apply the hyperplane projection algorithm (see [9] and [10] ). We consider three player’s game and let

m = 3

Table 1 shows that if we use the data from [1] in each period, then we get the same results.

Table 2 shows results for the case if elasticity parameter of first player increases up to 3.5 in three periods.

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.

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).

Finally, we assume that

γ_{i j} (k) = 1, j = 1, \dots, 4; k = 1, 2, 3

. 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

10000 - N_{j} (1) \times 3 = 8701

. If economic factor is equal to 3, then the number of uninsured people (which corresponds to inactive state) decreases down to

10000 - N_{j} (3) \times 3 = 163

. The results are presented in Table 5.

In this paper, we aim to investigate an extension of the one-period model in non-life insurance markets (cf. [1] ) 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 [11] and [12] ) 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.

Cite this paper
Battulga, G. , Altangerel, L. and Battur, G. (2018) An Extension of One-Period Nash Equilibrium Model in Non-Life Insurance Markets. Applied Mathematics, 9, 1339-1350. doi: 10.4236/am.2018.912087.

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