Bayesian Credibility Premium with GB2 Copulas

Bayesian shrinkage estimation is a familiar concept that arises from minimizing the Bayes risk under the squared error loss. For observations over a period of time, Bayesian shrinkage estimators may be used to predict the value of a response variable for a subject, given previously observed values. In this article, we formulate such estimators under a change of probability measure within the copula framework. Such reformulation is demonstrated using the multivariate generalized beta of the second kind (GB2) distribution. Within this family of GB2 copulas, we are able to derive explicit form of Bayesian shrinkage estimators. Numerical illustrations show the application of these estimators in determining experience-rated insurance premium. We consider generalized Pareto as a special case.


Introduction
For our purpose, a random variable is always well de ned on a given probability space and all random variables are continuous. Consider a sequence of random variables Y , . . . , Y T with the following Bayesian framework: • Y t |Θ ∼ f Yt (y t |θ) are independent for t = , . . . , T and • Θ ∼ p(θ) is the prior distribution. It is easy to see that the posterior density of Θ|Y T has the expression p(θ|Y T = y T ) = , (1.1) and the Bayesian posterior mean as E µ(Θ)|Y T = y T = µ(θ) p(θ|y T )dθ in the same manner as [4]. We obtain the following lemma. Lemma 1. Given the above Bayesian speci cation, the following relation holds: As a special case, let us assume the following model speci cation: ∼ N(Θ, ( − ρ)σ ) and Θ ∼ N(µ, ρ σ ).
Then Y T has a multivariate normal distribution with a common pairwise correlation of ρ so that we can write the correlation matrix as where T = ( , , . . . , ) is a row vector of 1's with dimension T and I T is a T × T identity matrix. In this case, we can show that all the marginals are also identically distributed as normal with mean µ and variance σ and the Bayesian posterior mean has the familiar form of a credibility premium, as shown in [2]: (1.4) where Y = T t= y t T, the average of past values.
The Bayesian speci cation captures the heterogeneity of each subject in a longitudinal dataset given observed values of y , . . . y T . The Bayesian posterior mean in equation (1.3) is the predicted value for the next time period T + given the historical observations. In the context of insurance, equation (1.4) is called a credibility-weighted premium and is used to calculate the subsequent year's premium given the past history of claims, for claims that follow the normal distribution.
This paper extends this result in the framework where we have a GB2 copula. In Section 2, we show a construction of the multivariate GB2 and study its properties. In Section 3, we brie y de ne the concept of copula and introduce the Bayesian credibility premium as an expectation under a change of probability measure. In Section 4, we show that using this change of measure, it becomes straightforward to derive Bayesian credibility premium with GB2 copulas. We consider a numerical illustration in Section 5. Section 6 discusses the generalized Pareto as a special case. Finally, we provide concluding remarks in Section 7.

The multivariate GB2 distribution
To construct the multivariate GB2 distribution, consider the following model speci cation: µ, p refers to a generalized inverse-gamma with density for p > . This speci cation can easily lead us to the following multivariate GB2 distribution by integrating out the scale parameter Θ: , and Ψ T = (ψ, ψ, . . . , ψ, k) is a vector of size (T + ) with equal rst T elements. Since Y t |Θ are i.i.d., then the marginal distribution is straightforward to determine by setting T = so that f t (y t ) = p y t B(ψ, k) c pk y pψ t y p t + c p ψ+k , where B(Ψ ) = B(ψ, k) with B(·, ·) is the beta function. For a random variable Y t with density given in equation (2.2), we can write Y t ∼ GB2(k, c, ψ, p). See [12] and [11] for applications of univariate GB2. A similar derivation of the multivariate GB2 has appeared in [17], but the parameterization is di erent from above. From the model speci cation, we can easily deduce the following moment properties: These properties lead us to the unconditional mean and variance, respectively, of a GB2 distribution: 3) The following lemma shows that the multivariate GB2 has a pairwise correlation structure.
Electronic copy available at: https://ssrn.com/abstract=3373377 Lemma 2. Suppose (Y , . . . , Y T ) follows a multivariate GB2 distribution as given in ( . ). Then they have a so-called pairwise correlation structure. In other words, and from equation (2.3), Since it is known that for any α, lim k→∞ Γ(k + α) Γ(k)k α = , we have that for any xed p, The following lemma shows that conditional distribution of each component of a multivariate GB2 random vector remains a member of the GB2 family.
Proof. The proof is straightforward by noting that

Change of probability measure with copulas
Copula is a widely used method to model dependency among multivariate observations. It has increased in popularity in recent years because of its widespread applications in several disciplines including, but not limited to, medical science, demography, hydrology, insurance, nance, and engineering. With copulas, one can decompose the marginal distributions and their dependence structure. See [9], [5], [6], [15], and [14].
Copulas are functions that join (or couple) the multivariate distribution functions to their onedimensional marginal distributions functions. See [8]. Speci cally, we have where F t (·) refers to the marginal distribution associated with Y t , H T is their joint distribution function, and C T is the corresponding copula function where subscripts of H and C refer to the dimension of the random vector. [16] proved the existence of copulas for every joint distribution function and demonstrated that they are indeed unique if the marginal distribution functions are continuous. It is also sometimes convenient to write this as Vectors shall be written in bold letters. For example, we shall denote the observed values of Y T by y T = (y , . . . , y T ) and similarly for Y T+ by y T+ = (y , . . . , y T , y T+ ) . For ease of notation, we will also denote the vectors u T = (u , . . . , u T ) and u T+ = (u , . . . , u T , u T+ ) . We shall assume that the densities of the copulas exist and are respectively denoted by and Notice that the marginal distribution functions have been denoted by and if the corresponding density functions exist, we denote them by Similarly, the multivariate density functions, if they exist, will be respectively denoted by In the following theorem, we derive the fundamental building block for deriving the Bayesian credibility premium, E [Y T+ |Y T = y T ], within a copula framework. Theorem 1. Consider the copula model satisfying the assumptions described in this section. The conditional expectation of Y T+ |Y T can be expressed in the following manner: where c T (u T ) and c T+ (u T+ ) are respectively de ned in (3.2) and (3.3), and that F T+ is a known marginal distribution function of Y T+ .
Proof. It is clear from the de nition of conditional density, that if it exists, we must have The numerator can be written as Electronic copy available at: https://ssrn.com/abstract=3373377 where u T+ is understood to be evaluated at the respective marginals F t (y t ) for t = , . . . , T, T + . Similarly, we have From these, we now have dy T+ , and the result given in (3.4) follows.
, we can verify that dF Q T+ = f Q T+ (y T+ ) · dy T+ becomes a probability measure because both c T (u T ) > and f T+ (y T+ ) > for any value of T. More speci cally, we have In general, we can derive the copula structure based on a random e ect framework as follows: where F Yt|θ and F t denotes the conditional and marginal distribution functions of Y t , respectively.

GB2 and Bayesian credibility premium
Note that when Y t ∼ GB2(k, c, ψ, p), we derive the marginal distribution F t as follows. By letting z = y p and v = z z + c p , we have Therefore, the GB2 copula can be derived in the same manner as follows:

1)
Electronic copy available at: https://ssrn.com/abstract=3373377 and its corresponding density is Note that if we substitute F t (y t ) for u t where F t (y t ) is a marginal distribution function of GB2, it turns out that c k,ψ (u , . . . , u T ) becomes h T (y T ), a joint density of multivariate GB2 distribution.
By Fubini's theorem, the bivariate GB2 copula is given as follows: where v i = (y i /c) p for i = , . From the above derivation, we see that the GB2 copula only depends on ψ and k, but not on c and p. Figure 1 provides a comparison of the contour plots of bivariate GB2 copulas using di erent set of parameters. Both parameters ψ and k describe the strength of the relationship as seen in this gure. For example, for a xed k, a higher value of ψ implies stronger dependence and for a xed ψ, a higher value of k implies weaker dependence. Now, by applying (4.2), we can get the following result which is a crucial step to derive the Bayesian credibility premium under the multivariate GB2 distribution model.
Proof. See appendix for details of the proof.
Based on Theorem 1, it is possible to evaluate the Bayesian credibility premium with GB2 copulas from the following theorem.
Theorem 2. Suppose (Y , . . . , Y T+ ) follows a multivariate distribution as described in (3.1) where C T is given as GB2 copula in (4.1) and the marginal distribution of Y t is given as F t . Then the Bayesian credibility premium is written as follows: , η T = T t= q t , and k T = ψT + k.
Proof. From Theorem 1 and (4.2), we can see that and F T+ (y T+ ) ∼ U[ , ], (4.4) can be expressed as follows: As a special case of Theorem 2, it is possible to derive a nice closed form of Bayesian credibility premium when Y T follows a multivariate GB2 distribution as follows.
Corollary 1. Suppose (Y , . . . , Y T+ ) follows a multivariate distribution as described in (3.1) where C T is given as GB2 copula in (4.1) and the marginal distribution of Y t is univariate GB2 distribution as in (2.2). Then the Bayesian credibility premium is given as follows: Proof. From Theorem 1 and Lemma 4, we can see that This ratio of densities of the copula, c T+ (u T+ ) c T (u T ) , in fact induces a change of probability measure so that in e ect, we can write the prediction as the following unconditional expectation under a change of measure dF Q T+ (y T+ ) = c T+ (u T+ ) c T (u T ) dF T+ (y T+ ). This change of measure allows us to construct an explicit expression for the posterior mean based on and 3). Therefore, we can get the following result directly from the de nition of c * T,p : If the marginal distribution of each does now follow GB2, then the integral in (4.3) might not able to be evaluated analytically. The corollary below provides a nice approximation of the credibility premium based on Monte Carlo method.

Corollary 2. A Monte Carlo approximation of Bayesian credibility premium in (4.3) is given as follows:
where u [m] for m = , . . . , M are random samples generated from U[ , ]. If T = , which means the case when there is no history of past claims, then η T = and k T = k so that (4.6) is reduced as follows: which is indeed a natural Monte Carlo approximation of the prior mean of Y T+ , E [Y T+ ].

Numerical illustration -insurance ratemaking
For numerical illustration, we consider the case where our primary variable of interest, y, is the amount of claim for a portfolio of insurance contracts. In particular, we have a set of random claims for T = periods: y , . . . , y . Determining the pure premium based on historical claim experience is the subject of experience rating and credibility. In this case, the pure premium is the Bayesian credibility premium with multivariate GB2 discussed in Corollary 1. This numerical example shows how the choice of parameters a ect Bayesian credibility premium with observed claim experience. To x ideas, we control the prior mean, µ, to be the same as 10 for all cases, but we vary the values of the corresponding coe cient of variation (CV), which is de ned Table 1 shows di erent combinations of parameters which have the same mean but with di erent coe cient of variations, respectively. . .
Using these combinations of parameters, we assume three scenarios of observed claims, where each scenario is represented by the randomly generated quantiles of incurred claims. The rst is a 'risky' scenario so that the average of generated quantiles is . %. The second is a 'normal' scenario so that the average of generated quantiles is . %. Finally, the last is a 'safe' scenario so that the average of generated quantiles is . %.
After assuming three scenarios based on the quantiles of the observed claims, we convert the quantile vectors to the observed claims under GB2 distribution for each set of parameters. We know that the 'weight factor' part of the GB2 credibility premium of the pure premium is given as + from Corollary 1. Therefore, the value of weight factor is determined both by the parameters and the observed claim values. Note that even though we have the same quantiles of generated claims, the generated claim amount can still vary along with the assumed set of parameters. Table 2 shows the result of the weight factors under all scenarios and parameter assumptions. Here, w H refers to the weight factors under the 'risky' scenario, w M to the weight factors under the 'normal' scenario, and w L to the weight factors under the 'safe' scenario, respectively. This numerical illustration presents some very intuitively interesting results. From Table 2, we can infer that as we have higher coe cient of variation, the impact of credibility weighing factor increases. Moreover, for a policyholder with relatively higher claim experience, the resulting credibility weighing factor is greater than 1, which implies a penalty to policyholders with unfavorable claim experience. On the other hand, for a policyholder with relatively favorable claim experience, the credibility weighing factor is less than 1, which implies a bonus to the policyholder with more favorable claim experience.

Special Case: Generalized Pareto
Generalized Pareto (GP) distribution is a special case of GB2 distribution when p = . We may derive the multivariate GP distribution based on the Bayesian speci cation in Section 2 with p = as follows: where • Gamma ψ, Θ ψ refers to a gamma with density and • I-Gamma(k, µ(k − )) refers to an inverse-gamma with density By integrating out Θ, we obtain the following multivariate GP distribution: For a random variable Y t with density given in equation (6.1), we can write Y t ∼ GP(k, c, ψ).
The following unconditional moments are straightforward to derive: By Lemma 2, we see that if (Y , . . . , Y T ) follows a multivariate GP distribution, then Again, we can check that lim k→∞ Cov(Y t , Y t ) = lim k→∞ ρ ,k,ψ = if t ≠ t . The parameter k gives a measure of the degree of correlation between pairs of GP random variables. Larger values of k imply less correlated variables. Under the GP model speci cation, from (4.5), the Bayesian posterior mean is given as which can be directly derived from the GB2 framework by letting p = . Interestingly, this has the form of a weighted average of prior mean µ and the average of previous observations, as shown in [1], [10], and [3]. Note that this is not at all surprising and is a natural result because it is well known that if Y t |Θ follows a distribution that belongs to the exponential family (Gamma distribution clearly belongs to the exponential family), then the posterior mean is exactly a linear combination of prior mean and sample mean of previous observations. For details of such result, see [7]. It is well known that gamma distribution is a choice for modeling the severity component of property and casualty insurance claims. The usual model speci cation entertained by many insurers is as follows: so that E [Yt] = µ and Var (Yt) = µ /ψ. We may regard this formulation as a limiting case of multivariate GP distribution because as k → ∞, Cov(Y t , Y t ) → and Θ → µ.
An insurance company may wish to use the multivariate GP distribution as a predictive claims model by carefully calibrating the value of k.

Conclusion
As stated in [13], the concept of Bayesian shrinkage estimation is not limited to a normally distributed random variable or a member of the exponential family. This article extends the literature by developing explicit forms of Bayesian credibility premium within the family of GB2 copulas. The development is based on a new concept of using change of probability measure for copulas; this result is stated in Theorem 1. This theorem is the fundamental foundation for developing the explicit forms. Such credibility premium are very useful in actuarial science and insurance for experience-based ratemaking, where contractholders may be rewarded or penalized depending on their own claim experience. The concepts in this paper can be readily applied and extended in several ways. First, note that it is possible to use regression function g(xβ) as prior mean, instead of the grand mean µ. Second, credibility premium with GB2 copulas, but with non-GB2 marginals, can be obtained so long as the random variables have continuous support as shown in Theorem 2 and Corollary 2. Finally, as in many diverse applications, the proposed credibility premium can have a wide ranging applications for various multiparameter inference and regression problems.  Electronic copy available at: https://ssrn.com/abstract=3373377