The premium of uncertain risk based on standard deviation principle

1 Introduction

Risk is ubiquitous in real life and it always leads to loss when judgments should be taken. Due to the complexity of the environment, one decision can result in many different outcomes. When potential losses are foreseen, the insured can transfer part of the risk to the insurance company by purchasing an insurance contract. Consequently, insurance is a risk transfer process in reality. That explains why the issue of insurance pricing has been in the spotlight for both policyholders and insurance companies. In general, the premium principle is to determine a non-random price for a risk variable with random values in a scientifically efficient way. One of the most prominent topics in actuarial science is the study of regularity in premium pricing.

Over the decades, many specific premium principles have been proposed in the stochastic case, such as the net premium principle, the expected utility premium principle, the variance-related premium principle, etc. Bühlmann [1] proposed zero utility premium principle based on the utility balance theory, but it does not fulfill the non-negative safe loading. Gerber [2] derived the Swiss premium principle, combining the equivalent utility theory with the zero utility principle. By modifying the square loss function to the exponential weighted loss function, Gerber [3] obtained the Esscher premium principle. Haezendonck and Goovaerts [4] raised normalized Young function and established Orlicz premium principle. Wang [5] included the distortion function into the premium pricing by employing Yaaer’s [6] dual theory of decision-making under risk. Young [7] studied eleven common premium principles on the basis of predecessors, and compared their respective properties.

Classical insurance pricing depends mainly on the expected utility theory, which plays an important role in actuarial science. To describe how to make a rational decision among indeterminate aspects, Neumann and Morgenstern [8] originated the axiomatic theory of expected utility. The above expected utility premium principle is based on the well-known expected utility theory. Since then, expected utility theory has been widely used to explain a wide range of economic events. Trowbridge [9] pointed out that expected utility theory is the cornerstone of actuarial science, so the decision makers regularly use it in scheme selection. Many scholars have analyzed economic behaviors in society by expected utility function such as Borch [10], Schoemaker [11], Rabin [12], and so on. Thereafter, risk theory and insurance developed at a rapid pace.

Zhang [13] put forward the empirical determination of the distorted risk premium in a Bayesian model and compared the convergence under different distortion functions. Through numerical simulations, Zhang et al. [14] compared the mean squared error of Bayesian estimation and reliability estimation under the variance premium principle. Du and Wen [15] advanced the generalized exponential premium principle and argued that nonparametric estimators perform better at small sample sizes. Zhang et al. [16] compared the efficiency of Bayesian estimation, reliability estimation, and maximum likelihood estimation to obtain an estimate of the risk premium when the prior distribution is not completely known. Zhang et al. [17] proposed an empirical Bayesian estimation method for the progress factor in the claim reserve model. Zhang [18] investigated the posterior determination of risk premium in the bivariate Bayesian aggregate risk model under the principle of variance-related premium. Zhang and Wen [19] studied parameter estimation under a two-parameter exponential distribution and considered quadratic Bayesian estimation to be better than classical reliability estimation.

In the past, risk measurement and insurance pricing were mostly handled by probability theory. However, in recent years, it has generally been treated by uncertainty theory, since risk indeterminacy can exhibit subjectivity at times. Uncertainty theory, established by Liu [20] in 2007 and redefined in 2009, is a whole new branch of mathematics that has found extensive applications in finance, insurance, and risk analysis. The advantage of addressing the subjectivity of risk is reflected in the application of empirical data from experts rather than previous historical data in uncertain statistics. Dong and Peng [21] put forward several distributions of insurance claimed under uncertain environment. Li et al. [22] proposed the uncertain net premium principle and extended it to the uncertain distortion premium principle, in order to improve the expected utility of the insured and the insurance company. Liu et al. [23] solved the premium constraints under uncertain random environment and proved the optimality of stop loss insurance. It has also been used successfully in dealing with numerous uncertain domains, such as mean-reverting currency model (Shen and Yao [24]), longevity risk securitization model (Gao and Liu [25]), insurance risk process with multiple claims (Liu and Yang [26]), and so on.

As uncertainty insurance continues to evolve, establishing a new uncertain premium principle within the existing framework is a critical topic. In this paper, the risk is defined as a non-negative uncertain variable and focus on the uncertain risk premium in the context of uncertainty theory. The rest of the paper is organized as follows. In the next section, several basic definitions and theorems on uncertain variable will be presented. In Section 3, we will introduce the uncertain variance-related premium principle and present some properties of risk under the uncertain standard deviation premium principle. In Section 4, the additional premium coefficient from the utility function will be derived, and two numerical examples of the maximum premium will be solved using the specified utility function and risk distribution. Then in Section 5, the unknown parameters of the policy with deductible will be estimated by uncertain moment estimation and uncertain maximum likelihood estimation, and the two uncertain premium values under different additional premium coefficients will be compared. Finally, there are some conclusions and future research directions in Section 6.

2 Preliminary

In this section, we will present several basic definitions and theorems on uncertain variable.

Definition 1. (Liu [20, 27]) Let L be a σ-algebra on a nonempty set Γ. A set function M : L → [ 0 , 1 ] is called an uncertain measure if it satisfies the following axioms:

Axiom 1: (Normality Axiom) For the universal set Γ, M { Γ } = 1 .

Axiom 2: (Duality Axiom) For any event Λ, M { Λ } + M { Λ c } = 1 .

Axiom 3: (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯, we have M { ⋃ i = 1 ∞ Λ i } ≤ ∑ i = 1 ∞ M { Λ i } . Axiom 4: (Product Axiom) Let ( Γ k , L k , M k ) be uncertainty spaces for k = 1, 2, ⋯. The product uncertain measure M is an uncertain measure satisfying M { ∏ k = 1 ∞ Λ k } = ⋀ k = 1 ∞ M k { Λ k } where Λ _k are arbitrarily chosen events from L k for k = 1, 2, ⋯, respectively.

Definition 2. (Liu [20]) The set function M is called an uncertain measure if it satisfies the four axioms.

Definition 3. (Liu [20]) An uncertain variable is a function ξ from an uncertainty space ( Γ , L , M ) to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B of real numbers.

Definition 4. (Liu [20]) Let ξ be an uncertain variable. Then its uncertainty distribution is defined by Φ ( x ) = M { ξ ≤ x } for any real number x.

Obviously, we have M { ξ > x } = 1 - Φ ( x ) .

Definition 5. (Liu [20]) Let ξ be an uncertain variable. The expected value of ξ is defined by E [ ξ ] = ∫ 0 + ∞ M { ξ ≥ r } d r - ∫ - ∞ 0 M { ξ ≤ r } d r provided that at least one of the two integrals is finite.

When the uncertainty distribution Φ of ξ is known, we have E [ ξ ] = ∫ 0 + ∞ [ 1 - Φ ( x ) ] d x - ∫ - ∞ 0 Φ ( x ) d x .

Theorem 1. (Liu [28]) Let ξ and η be independent uncertain variables with finite expected values. Then for any real numbers a and b, we have E [ a ξ + b η ] = aE [ ξ ] + bE [ η ] .

Definition 6. (Liu [20]) Let ξ be an uncertain variable with finite expected value e. Then the variance of ξ is V [ ξ ] = E [ ( ξ - e ) 2 ] .

On the basis of finite expected value e, if the uncertainty distribution Φ of ξ is known, then V [ ξ ] = ∫ 0 + ∞ [ 1 - Φ ( e + x ) + Φ ( e - x ) ] d x .

Theorem 2. (Liu [28]) If ξ is an uncertain variable with finite expected value, a and b are real numbers, then V [ a ξ + b ] = a 2 V [ ξ ] .

Example 1. An uncertain variable ξ is called linear if it has a linear uncertainty distribution Φ ( x ) = { 0 , if x ≤ a x - a b - a , if a < x ≤ b 1 , if x > b denoted by L ( a , b ) where a and b are real numbers with a < b.

According to Definition 5 and Definition 6, we have E [ ξ ] = a + b 2 , V [ ξ ] = ( b - a ) 2 12 .

Example 2. An uncertain variable ξ is called normal if it has a normal uncertainty distribution Φ ( x ) = ( 1 + exp ( π ( e - x ) 3 σ ) ) - 1 , x ∈ R denoted by N ( e , σ ) where e and σ are real numbers with σ > 0.

Then, its expected value and variance are E [ ξ ] = e , V [ ξ ] = σ 2 .

Example 3. An uncertain variable ξ is called zigzag if it has a zigzag uncertainty distribution Φ ( x ) = { 0 , if x ≤ a x - a 2 ( b - a ) , if a < x ≤ b x + c - 2 b 2 ( c - b ) , if b < x ≤ c 1 , if x > c denoted by Z ( a , b , c ) where a, b, c are real numbers with a < b < c .

It can be obtained through calculation that E [ ξ ] = a + 2 b + c 4 , V [ ξ ] = 5 a 2 + 4 b 2 + 5 c 2 - 4 ab - 4 bc - 6 ac 48 .

Definition 7. (Lio and Liu [33]) Let ξ₁, ξ₂, …, ξ _n be iid samples with uncertainty distribution F (z ∣  θ ) where θ is a set of unknown parameters, and let z₁, z₂, …, z _n be observed. The likelihood function is defined by L ( θ ∣ z 1 , z 2 , … , z n ) = lim Δ z → 0 ⋀ i = 1 n F ( z i + Δ z 2 ∣ θ ) - F ( z i - Δ z 2 ∣ θ ) Δ z .

3 Uncertain standard deviation premium principle

In this section, we will introduce the uncertain variance-related premium principle and present some properties of risk under the uncertain standard deviation premium principle.

Definition 8. Assume that the premium is a function P [X] defined on risk X, the uncertain variance-related premium principle P [ X ] = E [ X ] + g ( V [ X ] )(1) where monotone continuous increasing function g (x) : [0, + ∞) → [0, + ∞) and g (0) =0.

The principle of variance-related premium was proposed by Guerra and Centeno [29], which is widely used in actuarial science. By choosing different functions g (x), the uncertain variance-related premium principle can be transformed into different uncertain premium principles: if g (x) ≡0, it is the uncertain net premium principle; if g (x) = αx, it is the uncertain variance premium principle; if g ( x ) = α x , it is the uncertain standard deviation premium principle. Note that the expected value and variance are the first and second moments of the risk X. As the arithmetic root of variance, the standard deviation is also a first moment, so the uncertain standard deviation premium principle unifies the units. The rest of the paper is dominated by the uncertain standard deviation premium principle.

Definition 9. Suppose that ξ is a risk, then P [ ξ ] = E [ ξ ] + α V [ ξ ](2) is the uncertain standard deviation premium principle of ξ, where α > 0 is called the additional premium coefficient.

Example 4. Assume that η is a risk with uncertainty distribution ϒ (x) whose expected value is m. Under the proportional damage insurance, regard compensation amount γ = βη  (0 < β < 1) , the uncertainty distribution of γ is Ψ ( x ) = ϒ ( x β ) β ∈ ( 0 , 1 ) .

Thus, the premium of γ under uncertain standard deviation principle is P [ γ ] = E [ β η ] + α V [ β η ] = ∫ 0 + ∞ [ 1 - ϒ ( x β ) ] d x + α ∫ 0 + ∞ [ 1 - ϒ ( m + x β ] + ϒ ( m - x β ) ] d x = β m + α β ∫ 0 + ∞ [ 1 - ϒ ( m + x ) + ϒ ( m - x ) ] d x .(3) When calculating the premium of the uncertainty standard deviation principle, it is necessary to take into account not only the expected value of the risk, but also the standard deviation indicating the degree of risk volatility. There are two factors that affect the calculation of the premium, the additional premium coefficient and the actual risk distribution. Some properties of risk are listed and the premiums for various risks are compared.

Property 1. (Monotonicity) Let ξ and η be two risks with the same expected value, the insured with greater variance needs to pay more premiums.

Proof. Notice that the premium is determined jointly by the expected value and the standard deviation. On the premise of E (ξ) = E (η), if V (ξ) > V (η), then P (ξ) > P (η). Similarly, when V (ξ) < V (η), we can conclude P (ξ) < P (η).

This property means that the larger the risk dispersion at the same expected value of the two risks, the more the policyholder has to pay in premiums.

Property 2. (Risk Loading) For any risk ξ, P (ξ) ≥ E (ξ).

Proof. It is apparent that P [ξ] = E [ξ] + αV [ξ] ≥ E [ξ] for additional premium coefficient α > 0, the equal sign holds only if the variance is zero.

This property considers that the insurance company bears the additional expenses associated with insurance, so that the premium charged is always higher than the uncertain net premium. Otherwise, most insurance companies would be in bankruptcy.

Property 3. (Translation Invariance) For any risk ξ, P [ξ + k] = P [ξ] + k provided that t ≥ 0.

Proof. It can be confirmed by Theorem 1 that P [ ξ + k ] = E [ ξ + k ] + α V [ ξ + k ] = E [ ξ ] + α V [ ξ ] + k = P [ ξ ] + k .(4)

This property shows that when adding a constant k to uncertain risk ξ, the premium for ξ + k includes the constant k as well.

Property 4. (Positive Homogeneity) For any risk ξ, P [tξ] = tP [ξ] provided that t ≥ 0 is constant.

Proof. It can be confirmed by Theorem 2 that P [ t ξ ] = E [ t ξ ] + α V [ t ξ ] = tE [ ξ ] + α t 2 V [ ξ ] = tP [ ξ ] .(5)

This property indicates that the premium changes proportionally to the increase in risk. In other words, the premium for doubling a risk is twice as large as the premium for a single risk.

Property 5. (Independent additivity) Suppose that ξ and η are two independent risks with finite expected value and variance, we have P [ ξ + η ] = P [ ξ ] + P [ η ] .

Proof. For independent uncertain risk ξ and η with limited expected value and variance, then P [ ξ + η ] = E [ ξ + η ] + α V [ ξ + η ] = E [ ξ ] + E [ η ] + α V [ ξ ] + α V [ η ] = P [ ξ + η ] .(6)

This property employs that the sum of the premiums of two independent risks is equal to the sum of the premiums of both risks.

Property 6. (Continuity) Assume that ξ be an risk with uncertainty distribution Ψ (x). For any r ≥ 0, we have lim r → + ∞ P [ ξ ∧ r ] = P [ ξ ] .

Proof. The uncertainty distribution Φ (x) of ξ ∧ r is

Φ ( ξ ∧ r ) = { Ψ ( x ) , if 0 ≤ x < r 1 , if x ≥ r . Therefore, lim r → + ∞ P [ ξ ∧ r ] = lim r → + ∞ ∫ 0 r { ξ ≥ x } d x = P [ ξ ] .(7)

This property demonstrates that the approximation approach method can be used to estimate the theoretical premium of ξ.

In this section, under the uncertain standard deviation premium principle, we will define and solve the additional premium coefficient taking into account the risk-averse utility function. The similar definition in uncertain environment can be referred to Li and Peng [30], Zhou et al. [31], Chen and Park [32].

It is assumed that a risk-averse insured with the utility function u (x) has initial wealth w. Risk aversion is an attitude in which individuals prefer to accept a fixed amount of money rather than greater anticipated incomes with uncertain risk. Consider the instance of risk averter who is exposed to a potential risk Z, whose value ranges from 0 to w. The insured person will think about buying insurance to avoid the potential loss caused by the risk. As a result, the premium P ought to satisfy the following formula u ( w - P ) ≥ u ( w - Z ) . In the majority of cases, policyholders desire the lowest possible premium with little extra cost. Since u (x) is a non-decreasing continuous function, P _max is the maximum premium adequate to the insured. Furthermore, P _max is the solution of the implicit equation E [ u ( w - P max ) ] = E [ u ( w - Z ) ] .(8)

Generally speaking, more wealth will raise the utility level, so marginal utility function u′ (x) ≥0. The insured is risk-averse, which results in a decrease in the marginal utility, i.e., u^′′ (x) ≤0, and u (x) is a concave function. According to Jensen’s inequality, we know E [ u ( w - Z ) ] ≤ u ( E [ w - Z ] ) = u ( w - E [ Z ] )(9) for initial wealth w and loss caused by the risk Z. The decision makers are more willing to pay a fixed amount E [Z] rather than bear a risky loss Z, which indicates that they are risk averse.

Let e and v² be the expected value and variance of risk η. As a matter of fact, it is hard to get an accurate value for the maximum premium that policyholders can accept. Therefore, the maximum premium is estimated approximately by the Taylor expansion method.

For one thing, by the second terms in the series expansion of u (z) in w - e, we obtain u ( z ) ≈ u ( z 0 ) + u ′ ( z 0 ) ( z - z 0 ) + 1 2 u ′ ′ ( z 0 ) ( z - z 0 ) 2 where z₀ = w - e. Substituting w - η for z, we have u ( w - η ) ≈ u ( w - e ) + u ′ ( w - e ) ( e - η ) + 1 2 u ′ ′ ( w - e ) ( e - η ) 2 . Taking expected value on both sides of above the approximation yields, then E [ u ( w - η ) ] ≈ u ( w - e ) + 1 2 u ′ ′ ( w - e ) v 2 .(10)

For another, by the first terms in the series expansion of u (z) in (w - e) and substituting w - P _max for z, we have u ( w - P max ) ≈ u ( w - e ) + u ′ ( w - e ) ( e - P max ) .(11) Hence, it follows from the utility equilibrium equation E [u (w - η)] = u (w - P _max ) that P max ≈ e - 1 2 u ′ ′ ( w - e ) u ′ ( w - e ) v 2 .(12) According to (6) and (7), it is clear that P _max  ≥ e for given risk η.

Definition 10. Suppose that the risk averter has a utility function u (x), then u′ (x) >0textupandu^′′ (x) <0. The additional premium coefficient at initial wealth w and risk ξ is defined by α ( w - e ) = - 1 2 u ′ ′ ( w - e ) u ′ ( w - e ) v(13) where e and v are the expected value and the standard deviation of ξ.

The additional premium coefficient α could be used for measuring the risk dispersion and the decision makers’ degree of risk aversion. When the degree of risk aversion remains unchanged (i.e., when the utility function remains unchanged), the smaller the coefficient, the greater the degree of risk dispersion; when the degree of risk dispersion remains unchanged (i.e., the risk distribution function remains unchanged), the smaller the coefficient, the higher the degree of risk aversion represented by the corresponding utility function.

Based on the additional premium coefficient of the utility function u (x), then the maximum premium P _max to be paid for a risk ξ can be approximately rewritten as P max ≈ e + α ( w - e ) v .(14) When Equation (10) is established, the additional premium coefficient α is regarded as the maximum value of function α (x). Two numerical examples are given under the uncertain standard deviation premium principle, comparing the additional premium coefficient α of different risks under the same utility function with the additional premium coefficient α of the same risk under different utility functions.

Example 5. Suppose that a person with initial wealth 600 dollars facing potential risks ξ or η. ξ is normal uncertainty distribution N ( 100 , 5 ) and η is linear uncertainty distribution L ( 80 , 120 ) . The insured is a risk averse whose utility function is u ( x ) = x .

It is easy to obtain the expected value is e₁ = E (ξ) =100 and the standard deviation is v 1 = V ( ξ ) = 5 . The uncertainty distribution of η is derived, Φ ( x ) = { 0 , if x ≤ 80 x - 80 40 , if 80 < x ≤ 120 1 , if x < 120 . Thus, the expected value is e₂ = E (η) =100 and the standard deviation is v 2 = V ( η ) = 20 3 3 . The additional premium coefficient is α ( x ) = v 4 x , we calculated that α ( w - e 1 ) = 1 400 and α ( w - e 2 ) = 3 300 . Therefore, the maximum premium P max ( ξ ) = e 1 + α ( w - e 1 ) v 1 = 100.0125 P max ( η ) = e 2 + α ( w - e 2 ) v 2 = 100.0667 .

Example 6. Suppose that two risk-averse persons have the same initial wealth 600 dollars, and their utility functions are u₁ (x) = - x² + 1500x and u₂ (x) = - e^-0.01x. Assume that the insureds jointly facing with risk ζ is zigzag uncertainty distribution Z ( 80 , 95 , 130 ) , whose uncertainty distribution is Φ ( x ) = { 0 , if x ≤ 80 x - 80 30 , if 80 < x ≤ 95 x - 60 70 , if 95 < x ≤ 130 1 , if x > 130 Through calculation, we get the expected value is e = E (ζ) =100 and the standard deviation v = V ( ζ ) = 5 78 3 . The additional premium coefficients are α 1 ( w - e ) = v - 2 ( w - e ) + 1500 and α 2 ( w - e ) = v 200 , respectively. Hence, the maximum premium P 1 ( ζ ) = e + α 1 ( w - e ) v = 100.4333 P 2 ( ζ ) = e + α 2 ( w - e ) v = 101.0833 .

In this section, under the uncertain variance-related premium principle, two uncertain estimation methods will be used to estimate the unknown parameters in order to predict the future premium of a policy with deductible. There are many scholars who have studied parameter estimation in uncertain environment, see Lio and Liu [33], Zhang and Sheng [34], and Liu [35] for methods.

Deductible clauses are widely used in real life because they avoid high administrative costs for insurance companies caused by small claims and reduce premiums for policyholders. Suppose that X is an insurance policy with deductible b, the probable loss that the insurance company faces after underwriting the policy is given by Y = ( X - b ) + = { x - b , if x ≥ b 0 , if x < b . Assume that a group of claim amounts X _i   (i = 1, 2, ⋯ , n) are mutual independence and followed the lognormal uncertainty distribution LOGN ( μ , θ ) where μ and θ > 0 are unknown parameters. Under deductible clauses, the compensation amounts Y _i of the policy are observed as Y₁, Y₂, ⋯ , Y _n , then Y i = ( X i - b ) + i = 1 , 2 , ⋯ , n .(15) Thus, the uncertainty distribution of Y _i   (i = 1, 2, ⋯ , n) is obtained F ( y i ) = { F ( b ) , if y i = 0 F ( y i + b ) , if y i > 0 where F (x) is the lognormal uncertainty distribution. Under uncertain variance-related premium principle, predict the future premium through this formula P n + 1 = E [ Y n + 1 ] + g ( V [ Y n + 1 ] ) .(16)

Based on the samples Y _i   (i = 1, 2, ⋯ , n), two uncertain estimation methods are used to predict the future premium P_n+1. According to the uncertain moment estimation, the expected value of Y₁, Y₂, ⋯ , Y _n exists when standard deviation σ < π 3 , then { e ˆ = 1 n ∑ i = 1 n E [ Y i ] σ ˆ = 1 n ∑ i = 1 n V [ Y i ](17) the solution of Equation (14) is taken as the uncertain moment estimator. Therefore, the uncertain moment estimation of the future premium P_n+1 is P n + 1 ˆ M = e ˆ + g ( σ ˆ ) ,(18) where e ˆ = 1 n ∑ i = 1 n ∫ 0 + ∞ [ 1 - F ( y i ) ] d y i and σ ˆ = 1 n ∑ i = 1 n ∫ 0 + ∞ [ 1 - F ( e ˆ + y i ) + F ( e ˆ - y i ) ] d y i . Another method of estimating P_n+1 is based on uncertain maximum likelihood estimation.

Theorem 3. Let Y₁, Y₂, ⋯ , Y _n be independent identically distributed samples, and let y₁, y₂, ⋯ , y _n be observed. The lognormal uncertainty distribution F (y ∣ e, σ) is a continuous differentiable function where e and σ > 0 are unknown parameters. Under the deductible clauses with deductible b, the uncertain likelihood function is L ( e , σ ∣ y 1 + b , y 2 + b , ⋯ , y n + b ) = [ F ( b ) ] m π 3 σ exp ( π 3 σ ⋁ i = 1 n - m | e - ln ( y i + b ) | ) ⋁ i = 1 n - m y i [ 1 + exp ( π 3 σ ⋁ i = 1 n - m | e - ln ( y i + b ) | ) ] 2 . where indicative function m = ∑ i = 1 n I { y i = 0 } .

Proof. The general form of uncertain likelihood function is obtained by L ( θ ∣ l ( x 1 ) , l ( x 2 ) , ⋯ , l ( x n ) ) = ⋀ i = 1 n F ′ ( l ( x i ) ∣ θ ) . The lognormal uncertainty distribution LOGN ( e , σ ) is F ( y ∣ e , σ ) = [ 1 + exp ( π ( e - ln y ) 3 σ ) ] - 1 , y ∈ R . Given that y₁, y₂, ⋯ , y _n are observed, since F ′ ( y + b ∣ e , σ ) = π 3 σ exp ( π ( e - ln ( y + b ) ) 3 σ ) y [ 1 + exp ( π ( e - ln ( y + b ) ) 3 σ ) ] 2 , then we have L ( e , σ ∣ y 1 + b , y 2 + b , ⋯ , y n + b ) = ⋀ i = 1 n F ′ ( y i + b ∣ e , σ ) = [ F ( b ) ] m ⋀ i = 1 n - m F ′ ( y i + b ∣ e , σ ) = [ F ( b ) ] m ⋀ i = 1 n - m π 3 σ exp ( π ( e - ln ( y i + b ) ) 3 σ ) y i [ 1 + exp ( π ( e - ln ( y i + b ) ) 3 σ ) ] 2 . Furthermore, F′ (y + b ∣ e, σ) increased firstly and then decreased, reached its maximum value when ln(y _i  + b) = e  (i = 1, 2, ⋯ , n). Therefore, the uncertain likelihood function is L ( e , σ ∣ y 1 + b , y 2 + b , ⋯ , y n + b ) = [ F ( b ) ] m π 3 σ exp ( π 3 σ ⋁ i = 1 n - m | e - ln ( y i + b ) | ) ⋁ i = 1 n - m y i [ 1 + exp ( π 3 σ ⋁ i = 1 n - m | e - ln ( y i + b ) | ) ] 2 . This theorem is proved.

Theorem 4. Regard the uncertain maximum likelihood estimators as e^* and σ^*, the deductible clauses with deductible b have an uncertainty distribution F (y + b ∣ e^*, σ^*) when uncertain likelihood function L (e, σ ∣ y₁ + b, y₂ + b, ⋯ , y _n  + b) reaches its maximum value. The uncertain maximum likelihood estimators e^* and σ^* jointly solve the following problem, max e , σ > 0 L ( e , σ ∣ y 1 + b , y 2 + b , ⋯ , y n + b )(19) where L ( e , σ ∣ y 1 + b , y 2 + b , ⋯ , y n + b ) = [ F ( b ) ] m π 3 σ exp ( π 3 σ ⋁ i = 1 n - m | e - ln ( y i + b ) | ) ⋁ i = 1 n - m y i [ 1 + exp ( π 3 σ ⋁ i = 1 n - m | e - ln ( y i + b ) | ) ] 2 . Furthermore, e^* solves the minimization problem min e ⋁ i = 1 n - m | e - ln ( y i + b ) | , σ^* solves the maximization problem max σ > 0 [ F ( b ) ] m π 3 σ exp ( π 3 σ ⋁ i = 1 n - m | e * - ln ( y i + b ) | ) ⋁ i = 1 n - m y i [ 1 + exp ( π 3 σ ⋁ i = 1 n - m | e * - ln ( y i + b ) | ) ] 2 .

Proof. Now that the uncertain likelihood function L (e, σ ∣ y₁ + b, y₂ + b, ⋯ , y _n  + b) is decreasing with respect to ⋁ i = 1 n - m | e - ln ( y i + b ) | , then e^* solves the minimization problem min e ⋁ i = 1 n - m | e - ln ( y i + b ) | whose solution is e * = ⋀ i = 1 n - m ln ( y i + b ) + ⋁ i = 1 n - m ln ( y i + b ) 2 . After the uncertain maximum likelihood estimator e^* determining, the objective function for σ is uni-modal. Hence, σ^* solves the maximization problem max σ > 0 [ F ( b ) ] m π 3 σ exp ( π 3 σ ⋁ i = 1 n - m | e * - ln ( y i + b ) | ) ⋁ i = 1 n - m y i [ 1 + exp ( π 3 σ ⋁ i = 1 n - m | e * - ln ( y i + b ) | ) ] 2 . This theorem is proved.

It is simple to obtain the uncertain maximum likelihood estimation of the future premium P_n+1, then P n + 1 ˆ MLE = e * + g ( σ * ) .(20)

It follows from (14) that the uncertain moment estimator ( e ˆ , σ ˆ ) = ( 1.5197 , 1.5204 ) , then the calculation formula under uncertain moment estimation are P n + 1 ˆ M ( 1 ) = e ˆ + α σ ˆ 2 = 1.5197 + 2.3588 α P n + 1 ˆ M ( 2 ) = e ˆ + α σ ˆ = 1.5197 + 1.5204 α . To obtain the uncertain maximum likelihood estimators, the minimization problem should be solved firstly min e ⋁ i = 1 38 | e - ln ( y i + b ) | for acquiring e^* and max σ > 0 [ F ( b ) ] 12 π 3 σ exp ( π 3 σ ⋁ i = 1 38 | e * - ln ( y i + b ) | ) ⋁ i = 1 38 y i [ 1 + exp ( π 3 σ ⋁ i = 1 38 | e * - ln ( y i + b ) | ) ] 2 for acquiring σ^*. Thus, the uncertain maximum likelihood estimator (e^*, σ^*) = (1.4541, 0.5061) is obtained and the calculation formula under uncertain maximum likelihood estimation are P n + 1 ˆ MLE ( 1 ) = e * + α σ * 2 = 1.4486 + 0.2396 α P n + 1 ˆ MLE ( 2 ) = e * + α σ * = 1.4486 + 0.4895 α .

From the above selection of g (x) that g₁ (x) and g₂ (x) correspond to the uncertain variance premium principle and the uncertain standard deviation premium principle, respectively. For different additional premium coefficients α, Table 2 can be acquired by calculating the premium values under the two uncertain estimation methods.

Uncertain moment estimation is simpler to utilize, but it is prone to extreme values in small sample numbers, resulting in considerable deviations in calculated results. Although the computation of the uncertain maximum likelihood estimation is slightly more involved, the estimated variance is closer to the original value. When the two estimation methods are compared, it is discovered that the rate of change of the premium calculated by the uncertain maximum likelihood estimation is more steady. By comparing the two premium principles, it can be seen that the uncertain standard deviation premium principle is less affected by risk fluctuations, reducing the price sensitivity of risk-averse persons.

In this paper, we discuss the risk premium within the framework of uncertainty theory. The uncertain variance-related premium principle is proposed, and the uncertain standard deviation premium principle is further studied, and some properties of risk are given. Based on the risk-averse utility function, the formula of additional premium coefficient is advanced. The specified utility function and risk distribution are used to solve two numerical examples of the maximum premium. Uncertain moment estimation and uncertain maximum likelihood estimation are first introduced in premium prediction to estimate the unknown parameters of the risk distribution. When estimating policies with deductibles, the uncertain maximum likelihood estimation is less susceptible to extreme values and the rate of change in premiums is more stable. Moreover, the uncertain standard deviation premium principle is less affected by risk fluctuations, which can reduce the price sensitivity of the policyholders. Future research may consider the establishment of other uncertain premium principles that have not yet been proposed, compare them with existing uncertain premium principles, and find their respective applicable insurance types.

Acknowledgments

This work was funded by the Key Research and Development Plan Project of Xinjiang Uygur Autonomous Region (Grant No. 2021B03003-1) and the National Natural Science Foundation of China (Grant No. 12061072).