# Multivariate Frequency-Severity Regression Models in Insurance

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## Abstract

**:**

## 1. Introduction and Motivation

- N, the number of claims (events),
- ${y}_{k},\phantom{\rule{3.33333pt}{0ex}}k=1,...,N,\phantom{\rule{3.33333pt}{0ex}}$ the amount of each claim (loss), and
- $S={y}_{1}+\cdots +{y}_{N},$ the aggregate claim amount.

**Importance of Modeling Frequency.**The aggregate claim amount S is the key element for an insurer’s balance sheet, as it represents the amount of money paid on claims. So, why do insurance companies regularly track the frequency of claims as well as the claim amounts? As in an earlier review [1], we can segment these reasons into four categories: (i) features of contracts; (ii) policyholder behavior and risk mitigation; (iii) databases that insurers maintain; and (iv) regulatory requirements.

- Contractually, it is common for insurers to impose deductibles and policy limits on a per occurrence and on a per contract basis. Knowing only the aggregate claim amount for each policy limits any insights one can get into the impact of these contract features.
- Covariates that help explain insurance outcomes can differ dramatically between frequency and severity. For example, in healthcare, the decision to utilize healthcare by individuals (the frequency) is related primarily to personal characteristics whereas the cost per user (the severity) may be more related to characteristics of the healthcare provider (such as the physician). Covariates may also be used to represent risk mitigation activities whose impact varies by frequency and severity. For example, in fire insurance, lightning rods help to prevent an accident (frequency) whereas fire extinguishers help to reduce the impact of damage (severity).
- Many insurers keep data files that suggest developing separate frequency and severity models. For example, insurers maintain a “policyholder” file that is established when a policy is written. A separate file, often known as the “claims” file, records details of the claim against the insurer, including the amount. These separate databases facilitate separate modeling of frequency and severity.
- Insurance is a closely monitored industry sector. Regulators routinely require the reporting of claims numbers as well as amounts. Moreover, insurers often utilize different administrative systems for handling small, frequently occurring, reimbursable losses, e.g., prescription drugs, versus rare occurrence, high impact events, e.g., inland marine. Every insurance claim means that the insurer incurs additional expenses suggesting that claims frequency is an important determinant of expenses.

**Importance of Including Covariates.**In this work, we assume that the interest is in the joint modeling of frequency and severity of claims. In actuarial science, there is a long history of studying frequency, severity and the aggregate claim for homogeneous portfolios; that is, identically and independently distributed realizations of random variables. See any introductory actuarial text, such as [2], for an introduction to this rich literature.

**Importance of Multivariate Modeling.**To summarize reasons for examining insurance outcomes on a multivariate basis, we utilize an earlier review in [4]. In that paper, frequencies were restricted to binary outcomes, corresponding to a claim or no claim, known as “two-part” modeling. In contrast, this paper describes more general frequency modeling, although the motivation for examining multivariate outcomes are similar. Analysts and managers gain useful insights by studying the joint behavior of insurance risks, i.e., a multivariate approach:

- For some products, insurers must track payments separately by component to meet contractual obligations. For example, in motor vehicle coverage, deductibles and limits depend on the coverage type, e.g., bodily injury, damage to one’s own vehicle, or damage to another party; is natural for the insurer to track claims by coverage type.
- For other products, there may be no contractual reasons to decompose an obligation by components and yet the insurer does so to help better understand the overall risk. For example, many insurers interested in pricing homeowners insurance are now decomposing the risk by “peril”, or cause of loss. Homeowners is typically sold as an all-risk policy, which covers all causes of loss except those specifically excluded. By decomposing losses into homogenous categories of risk, actuaries seek to get a better understanding of the determinants of each component, resulting in a better overall predictor of losses.
- It is natural to follow the experience of a policyholder over time, resulting in a vector of observations for each policyholder. This special case of multivariate analysis is known as “panel data”, see, for example, [5].
- In the same fashion, policy experience can be organized through other hierarchies. For example, it is common to organize experience geographically and analyze spatial relationships.
- Multivariate models in insurance need not be restricted to only insurance losses. For example, a study of term and whole life insurance ownership is in [6]. As an example in customer retention, both [7,8] advocate for putting the customer at the center of the analysis, meaning that we need to think about the several products that a customer owns simultaneously.

**Dependence and Contagion**. We have seen in the above discussion that dependencies arise naturally when modeling insurance data. As a first approximation, we typically think about risks in a portfolio as being independent from one another and rely upon risk pooling to diversify portfolio risk. However, in some cases, risks share common elements such as an epidemic in a population, a natural disaster such as a hurricane that affects many policyholders simultaneously, or an interest rate environment shared by policies with investment elements. These common (pandemic) elements, often known as “contagion”, induce dependencies that can affect a portfolio’s distribution significantly.

- Dependencies may impact the statistical significance of parameter estimates.
- When we examine the distribution of one variable conditional on another, dependencies are important.
- For prediction, the degree of dependency affects the degree of reliability of our predictions.
- Insurers want to construct products that do not expose them to extreme variation. They want to understand the distribution of a product that has many identifiable components; to understand the distribution of the overall product, one strategy is to describe the distribution of each product and a relationship among the distributions.

**Plan for the Paper**. The following is a plan to introduce readers further to the topic. Section 2 gives a brief overview of univariate models, that is, regression models with a single outcome for a response. This section sets the tone and notation for the rest of the paper. Section 3 provides an overview of multivariate modeling, focusing on the “copula” regression approach described here. This section discusses continuous, discrete, and mixed (Tweedie) outcomes. For our regression applications, the focus is mainly on a family of copulas known as “elliptical”, because of their flexibility of modeling pairwise dependence and wide usages in multivariate analysis. Section 3 also summarizes a modeling strategy for the empirical approach of copula regression.

## 2. Univariate Foundations

#### 2.1. Frequency-Severity

#### 2.2. Modeling Frequency Using GLMs

**β**is a vector of parameters associated with the covariates. This function is used because it yields desirable parameter interpretations, seems to fit data reasonably well, and ties well with other approaches traditionally used in actuarial ratemaking applications [12].

#### 2.3. Modeling Severity

**Modeling Severity Using GLMs**. For insurance analysts, one strength of the GLM approach is that the same set of routines can be used for continuous as well as discrete outcomes. For severities, it is common to use a gamma or inverse Gaussian distribution, often with a logarithmic link (primarily for parameter interpretability).

**Modeling Severity Using**GB2. The GLM is the workhorse for industry analysts interested in analyzing the severity of claims. Naturally, because of the importance of claims severity, a number of alternative approaches have been explored, cf., [13] for an introduction. In this review, we focus on a specific alternative, using a distribution family known as the “generalized beta of the second kind”, or GB2, for short.

#### 2.4. Tweedie Model

**β**, as the latter is a symbol we will use for the coefficients corresponding to explanatory variables. Then, $S={y}_{1}+\dots +{y}_{N}$ is a Poisson sum of gammas.

## 3. Multivariate Models and Methods

#### 3.1. Copula Regression

**Introducing Copulas.**Specifically, a copula is a multivariate distribution function with uniform marginals. Let ${U}_{1},\dots ,{U}_{p}$ be p uniform random variables on $(0,1)$. Their joint distribution function

**Regression with Copulas.**In a regression context, we assume that there are covariates $\mathbf{x}$ associated with outcomes $\mathbf{y}={({y}_{1},\dots ,{y}_{p})}^{\prime}$. In a parametric context, we can incorporate covariates by allowing them to be functions of the distributional parameters.

#### 3.2. Multivariate Severity

**α**. With this, using Equation (4), the log-likelihood function of the ith risk is written as

#### 3.3. Multivariate Frequency

#### 3.4. Multivariate Tweedie

#### 3.5. Association Structures and Elliptical Copulas

- The multivariate normal (Gaussian) distribution has provided a foundation for multivariate data analysis, including regression. By permitting a flexible structure for the mean, one can readily incorporate complex mean structures including high order polynomials, categorical variables, interactions, semi-parametric additive structures, and so forth. Moreover, the variance structure readily permits incorporating time series patterns in panel data, variance components in longitudinal data, spatial patterns, and so forth. One way to get a feel for the breadth of variance structures readily accommodated is to examine options in standard statistical software packages such as
`PROC Mixed`in [37] (for example, the`TYPE`switch in the`RANDOM`statement permits the choice of over 40 variance patterns). - In many applications, appropriately modeling the mean and second moment structure (variances and covariances) suffices. However, for other applications, it is important to recognize the underlying outcome distribution and this is where copulas come into play. As we have seen, copulas are available for any distribution function and thus readily accommodate binary, count, and long-tail distributions that cannot be adequately approximated with a normal distribution. Moreover, marginal distributions need not be the same, e.g., the first outcome may be a count Poisson distribution and the second may be a long-tail gamma.
- Pair copulas (cf., [25]) may well represent the next step in the evolution of regression modeling. A copula imposes the same dependence structure on all p outcomes whereas a pair copula has the flexibility to allow the dependence structure itself to vary in a disciplined way. This is done by focusing on the relationship between pairs of outcomes and examining conditional structures to form the dependence of the entire vector of outcomes. This approach is useful for high dimensional outcomes (where p is large), an important developing area of statistics. This represents an excellent future step in copula regression modeling that is not addressed further in this article.

#### 3.6. Assessing Dependence

- Copula identification begins after marginal models have been fit. Then, use the “Cox-Snell” residuals from these models to check for association. Create simple correlation statistics (Spearman, polychoric) as well as plots ($pp$ and tail dependence plots) to look for dependence structures and identify a parametric copula.
- After a model identification, estimate the model and examine how well the model fits. Examine the residuals to search for additional patterns using, for example, correlation statistics and t-plot (for elliptical copulas). Examine the statistical significance of fitted association parameters to seek a simpler fit that captures the important tendencies of the data.
- Compare the fitted model to alternatives. Use overall goodness of fit statistics for comparisons, including AIC and BIC, as well as cross-validation techniques. For nested models, compare via the likelihood ratio test and use Vuong’s procedure for comparing non-nested alternative specifications.
- Compare the models based on a held-out sample. Use statistical measures and economically meaningful alternatives such as the Gini statistic.

#### 3.7. Frequency-Severity Modeling Strategy

- Fit the mean structure. Historically, this is the most important aspect. One can apply robust standard error procedures to get consistent and approximately normally distributed coefficients, assuming a correct mean structure.
- Fit the variance structure with a selected distribution. In GLMs, the choice of the distribution dictates the variance structure that can be over-ruled with a separately specified variance, e.g., a “double GLM.”
- Fit the dependence structure with a choice of copula.

#### 3.7.1. Identification and Estimation

#### 3.7.2. Model Validation

## 4. Frequency Severity Dependency Models

## 5. LGPIF Case Study

#### 5.1. Data / Problem Description

#### Explanatory Variables

#### 5.2. Marginal Model Fitting—Zero/One Frequency, GB2 Severity

#### 5.2.1. BC (Building and Contents) Frequency Modeling

#### 5.2.2. BC (Building and Contents) Severity Modeling

#### 5.2.3. Building and Contents Model Summary

#### 5.2.4. Marginal Models for Other Lines

#### 5.3. Copula Identification and Fitting

#### 5.3.1. Frequency Severity Dependence

#### 5.3.2. Dependence between Different Lines

## 6. Out-of-Sample Validation

#### 6.1. Spearman Correlation

#### 6.2. Gini Index

## 7. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## A. Appendix

#### A.1. Alternative Way of Choosing Location Parameters for GB2

#### A.2. Other Lines

#### A.2.1. IM (Contractor’s Equipment)

Empical | ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB | |
---|---|---|---|---|---|---|---|

0 | 4386 | $4381.660$ | $4383.572$ | $4351.363$ | $4383.527$ | $4384.718$ | $4384.736$ |

1 | 182 | $189.282$ | $184.517$ | $233.111$ | $191.214$ | $188.278$ | $188.252$ |

2 | 40 | $35.986$ | $37.159$ | $29.992$ | $31.187$ | $32.558$ | $32.560$ |

3 | 6 | $10.383$ | $11.428$ | $5.794$ | $9.313$ | $9.716$ | $9.719$ |

4 | 4 | $3.237$ | $3.662$ | $1.311$ | $3.555$ | $3.668$ | $3.669$ |

5 | 2 | $1.009$ | $1.155$ | $0.324$ | $1.548$ | $1.564$ | $1.565$ |

6 | 2 | $0.311$ | $0.357$ | $0.081$ | $0.740$ | $0.724$ | $0.724$ |

≥7 | 0 | $0.132$ | $0.151$ | $0.024$ | $0.889$ | $0.763$ | $0.764$ |

0 proportion | $0.949$ | $0.948$ | $0.948$ | $0.941$ | $0.948$ | $0.949$ | $0.949$ |

1 proportion | $0.039$ | $0.041$ | $0.040$ | $0.050$ | $0.041$ | $0.041$ | $0.041$ |

ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB |
---|---|---|---|---|---|

13.046 | 11.204 | 74.788 | 7.335 | 6.497 | 6.493 |

#### A.2.2. PN (Comprehensive New)

Empical | ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB | |
---|---|---|---|---|---|---|---|

0 | 1323 | $1318.274$ | $1310.359$ | $1260.404$ | $1324.292$ | $1327.832$ | $1323.999$ |

1 | 154 | $116.320$ | $161.105$ | $153.445$ | $148.051$ | $141.023$ | $149.927$ |

2 | 50 | $53.333$ | $28.759$ | $78.837$ | $52.433$ | $53.831$ | $50.552$ |

3 | 33 | $49.295$ | $30.636$ | $62.798$ | $32.670$ | $33.472$ | $32.031$ |

4 | 19 | $41.465$ | $32.238$ | $41.674$ | $22.728$ | $23.348$ | $22.648$ |

5 | 16 | $28.661$ | $28.095$ | $22.970$ | $16.145$ | $16.650$ | $16.341$ |

6 | 13 | $16.614$ | $20.560$ | $10.853$ | $11.546$ | $11.920$ | $11.827$ |

7 | 7 | $8.282$ | $12.953$ | $4.510$ | $8.291$ | $8.537$ | $8.557$ |

8 | 4 | $3.623$ | $7.169$ | $1.683$ | $5.972$ | $6.111$ | $6.185$ |

9 | 4 | $1.413$ | $3.540$ | $0.573$ | $4.314$ | $4.371$ | $4.466$ |

10 | 3 | $0.497$ | $1.579$ | $0.180$ | $3.124$ | $3.124$ | $3.221$ |

11 | 1 | $0.159$ | $0.643$ | $0.053$ | $2.267$ | $2.232$ | $2.320$ |

12 | 2 | $0.047$ | $0.241$ | $0.015$ | $1.649$ | $1.593$ | $1.670$ |

13 | 4 | $0.013$ | $0.084$ | $0.004$ | $1.202$ | $1.137$ | $1.201$ |

14 | 2 | $0.003$ | $0.027$ | $0.001$ | $0.878$ | $0.811$ | $0.863$ |

15 | 1 | $0.001$ | $0.008$ | 0 | $0.643$ | $0.578$ | $0.620$ |

16 | 1 | 0 | $0.002$ | 0 | $0.471$ | $0.412$ | $0.445$ |

≥17 | 1 | 0 | $0.001$ | 0 | $0.788$ | $0.651$ | $0.712$ |

0 proportion | $0.808$ | $0.805$ | $0.800$ | $0.769$ | $0.809$ | $0.811$ | $0.809$ |

1 proportion | $0.094$ | $0.071$ | $0.098$ | $0.094$ | $0.090$ | $0.086$ | $0.092$ |

ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB |
---|---|---|---|---|---|

31,776.507 | 2113.085 | 93,179.199 | 11.609 | 14.537 | 11.853 |

#### A.2.3. PO (Comprehensive Old)

Empical | ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB | |
---|---|---|---|---|---|---|---|

0 | 1875 | $1873.859$ | $1867.234$ | $1811.952$ | $1879.908$ | $1880.884$ | $1879.944$ |

1 | 155 | $121.646$ | $153.552$ | $180.597$ | $142.801$ | $139.962$ | $144.827$ |

2 | 42 | $54.712$ | $34.118$ | $76.153$ | $45.728$ | $45.669$ | $43.381$ |

3 | 26 | $38.944$ | $28.731$ | $40.240$ | $24.579$ | $24.945$ | $23.977$ |

4 | 12 | $24.618$ | $21.921$ | $18.234$ | $15.068$ | $15.469$ | $15.010$ |

5 | 8 | $13.332$ | $14.536$ | $7.152$ | $9.671$ | $10.002$ | $9.780$ |

6 | 7 | $6.366$ | $8.598$ | $2.512$ | $6.361$ | $6.609$ | $6.509$ |

7 | 4 | $2.757$ | $4.662$ | $0.814$ | $4.256$ | $4.434$ | $4.397$ |

8 | 2 | $1.109$ | $2.380$ | $0.248$ | $2.886$ | $3.010$ | $3.006$ |

9 | 2 | $0.421$ | $1.173$ | $0.072$ | $1.979$ | $2.065$ | $2.076$ |

10 | 1 | $0.153$ | $0.570$ | $0.020$ | $1.371$ | $1.429$ | $1.447$ |

11 | 1 | $0.054$ | $0.276$ | $0.005$ | $0.958$ | $0.998$ | $1.017$ |

≥12 | 3 | $0.026$ | $0.250$ | $0.002$ | $2.339$ | $2.426$ | $2.521$ |

0 proportion | $0.877$ | $0.876$ | $0.873$ | $0.847$ | $0.879$ | $0.880$ | $0.879$ |

1 proportion | $0.072$ | $0.057$ | $0.072$ | $0.084$ | $0.067$ | $0.065$ | $0.068$ |

ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB |
---|---|---|---|---|---|

387.671 | 43.127 | 5365.758 | 2.995 | 3.824 | 2.512 |

#### A.2.4. CN (Collision New)

Empical | ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB | |
---|---|---|---|---|---|---|---|

0 | 1159 | $1157.476$ | $1153.549$ | $1090.320$ | $1168.332$ | $1169.317$ | $1163.521$ |

1 | 228 | $201.586$ | $229.228$ | $274.394$ | $210.934$ | $209.282$ | $226.545$ |

2 | 74 | $79.485$ | $56.894$ | $95.634$ | $69.231$ | $69.270$ | $60.183$ |

3 | 26 | $43.915$ | $36.167$ | $40.251$ | $33.137$ | $33.325$ | $30.364$ |

4 | 16 | $24.613$ | $23.909$ | $17.118$ | $18.478$ | $18.644$ | $17.630$ |

5 | 9 | $12.536$ | $14.538$ | $6.987$ | $10.951$ | $11.066$ | $10.846$ |

6 | 3 | $5.718$ | $7.937$ | $2.723$ | $6.677$ | $6.749$ | $6.866$ |

7 | 3 | $2.352$ | $3.904$ | $1.018$ | $4.137$ | $4.178$ | $4.422$ |

8 | 4 | $0.881$ | $1.745$ | $0.366$ | $2.589$ | $2.612$ | $2.881$ |

9 | 1 | $0.303$ | $0.716$ | $0.127$ | $1.633$ | $1.645$ | $1.895$ |

10 | 0 | $0.097$ | $0.272$ | $0.042$ | $1.036$ | $1.043$ | $1.255$ |

11 | 3 | $0.029$ | $0.096$ | $0.014$ | $0.661$ | $0.664$ | $0.837$ |

12 | 0 | $0.008$ | $0.032$ | $0.004$ | $0.424$ | $0.425$ | $0.561$ |

13 | 1 | $0.002$ | $0.010$ | $0.001$ | $0.273$ | $0.273$ | $0.378$ |

14 | 1 | $0.001$ | $0.003$ | 0 | $0.176$ | $0.176$ | $0.256$ |

15 | 1 | 0 | $0.001$ | 0 | $0.114$ | $0.114$ | $0.174$ |

≥16 | 0 | 0 | 0 | 0 | $0.176$ | $0.175$ | $0.296$ |

0 proportion | $0.758$ | $0.757$ | $0.754$ | $0.713$ | $0.764$ | $0.765$ | $0.761$ |

1 proportion | $0.149$ | $0.132$ | $0.150$ | $0.179$ | $0.138$ | $0.137$ | $0.148$ |

ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB |
---|---|---|---|---|---|

10,932.035 | 1791.868 | 15,221.056 | 29.911 | 30.378 | 22.574 |

#### A.2.5. CO (Collision, Old)

Empical | ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB | |
---|---|---|---|---|---|---|---|

0 | 1651 | $1649.151$ | $1647.342$ | $1600.173$ | $1654.590$ | $1656.293$ | $1653.325$ |

1 | 224 | $197.854$ | $220.281$ | $262.576$ | $218.632$ | $212.240$ | $220.052$ |

2 | 63 | $85.200$ | $63.848$ | $84.408$ | $66.972$ | $70.829$ | $67.211$ |

3 | 34 | $42.854$ | $37.325$ | $37.334$ | $30.322$ | $32.286$ | $31.127$ |

4 | 22 | $21.236$ | $21.693$ | $16.911$ | $16.287$ | $16.989$ | $16.606$ |

5 | 5 | $9.765$ | $11.729$ | $7.164$ | $9.512$ | $9.603$ | $9.511$ |

6 | 2 | $4.137$ | $5.815$ | $2.811$ | $5.824$ | $5.650$ | $5.671$ |

7 | 5 | $1.647$ | $2.658$ | $1.040$ | $3.673$ | $3.408$ | $3.468$ |

8 | 3 | $0.645$ | $1.144$ | $0.373$ | $2.365$ | $2.091$ | $2.158$ |

9 | 3 | $0.265$ | $0.488$ | $0.133$ | $1.547$ | $1.301$ | $1.361$ |

≥10 | 1 | $0.230$ | $0.516$ | $0.076$ | $2.495$ | $1.883$ | $2.024$ |

0 proportion | $0.820$ | $0.819$ | $0.818$ | $0.795$ | $0.822$ | $0.823$ | $0.822$ |

1 proportion | $0.111$ | $0.098$ | $0.109$ | $0.130$ | $0.109$ | $0.105$ | $0.109$ |

ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB |
---|---|---|---|---|---|

60.691 | 25.206 | 121.987 | 10.387 | 11.440 | 10.370 |

#### A.3. Tweedie Margins

Variable Name | BC | IM | PN | ||||||
---|---|---|---|---|---|---|---|---|---|

Estimate | Standard Error | Estimate | Standard Error | Estimate | Standard Error | ||||

(Intercept) | 5.855 | 0.969 | *** | 8.404 | 1.081 | *** | 6.284 | 0.437 | *** |

lnCoverage | 0.758 | 0.155 | *** | 1.022 | 0.134 | *** | 0.395 | 0.107 | *** |

lnDeduct | 0.147 | 0.148 | −0.277 | 0.154 | . | ||||

NoClaimCredit | −0.272 | 0.371 | −0.330 | 0.244 | −0.570 | 0.296 | . | ||

EntityType: City | 0.264 | 0.574 | 0.223 | 0.406 | 0.930 | 0.497 | . | ||

EntityType: County | 0.204 | 0.719 | 0.671 | 0.501 | 2.550 | 0.462 | *** | ||

EntityType: Misc | −0.380 | 0.729 | −1.945 | 1.098 | . | −0.010 | 0.942 | ||

EntityType: School | 0.072 | 0.521 | −0.340 | 0.520 | 0.036 | 0.474 | |||

EntityType: Town | 0.940 | 0.658 | −0.487 | 0.476 | 0.185 | 0.586 | |||

$\varphi $ | 165.814 | 849.530 | 376.190 | ||||||

P | 1.669 | 1.461 | 1.418 | ||||||

Variable Name | PO | CN | CO | ||||||

Estimate Error | Standard | Estimate Error | Standard | Estimate Error | Standard | ||||

(Intercept) | 5.868 | 0.489 | *** | 8.263 | 0.294 | *** | 7.889 | 0.340 | *** |

lnCoverage | 0.860 | 0.119 | *** | 0.474 | 0.098 | *** | 0.841 | 0.117 | *** |

lnDeduct | |||||||||

NoClaimCredit | 0.155 | 0.319 | −0.369 | 0.253 | −1.025 | 0.331 | ** | ||

EntityType: City | 0.747 | 0.612 | 0.169 | 0.347 | −0.723 | 0.540 | |||

EntityType: County | 1.414 | 0.577 | * | 1.112 | 0.325 | *** | 0.863 | 0.434 | * |

EntityType: Misc | 0.033 | 0.925 | −0.596 | 0.744 | −0.579 | 0.939 | |||

EntityType: School | 0.989 | 0.544 | . | −0.631 | 0.316 | * | 0.477 | 0.399 | |

EntityType: Town | −2.482 | 1.123 | * | −1.537 | 0.499 | ** | −0.628 | 0.564 | |

$\varphi $ | 322.662 | 336.297 | 302.556 | ||||||

P | 1.508 | 1.467 | 1.527 |

#### A.4. Dependence of Frequency and Severity

#### A.4.1. Moments

#### A.4.2. Average Severity

#### A.4.3. Correlation and Dependence

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**Figure 2.**Out-of-Sample Validation for Independent Tweedie. (In these plots, the conditional mean for each policyholder is plotted against the claims.)

**Figure 3.**Out-of-Sample Validation for Dependent Frequency-Severity. (In these plots, the claim scores for each line is simulated from the frequency-severity model with dependence, using a Monte Carlo approach with B = 50,000 samples from the normal copula. The model with 01-NB and GB2 marginals show clear improvement for the BC line, in particular for the upper tail prediction. For other lines such as CO, the GB2 marginal results in miss-scaling).

Average Frequency | Average Severity | Annual Claims in Each Year | Average Coverage (Million) | Number of Claims | Number of Observations | |
---|---|---|---|---|---|---|

BC | 0.879 | 9868 | 17,143 | 37.050 | 4992 | 5660 |

IM | 0.056 | 624 | 766 | 0.848 | 318 | 4622 |

PN | 0.159 | 197 | 466 | 0.158 | 902 | 1638 |

PO | 0.103 | 311 | 504 | 0.614 | 587 | 2138 |

CN | 0.127 | 374 | 744 | 0.096 | 720 | 1529 |

CO | 0.120 | 538 | 951 | 0.305 | 680 | 2013 |

Average Frequency | Average Severity | Annual Claims in Each Year | Average Coverage (Million) | Number of Claims | Number of Observations | |
---|---|---|---|---|---|---|

BC | 0.945 | 8352 | 20,334 | 42.348 | 1038 | 1095 |

IM | 0.076 | 382 | 645 | 0.972 | 83 | 904 |

PN | 0.224 | 307 | 634 | 0.172 | 246 | 287 |

PO | 0.128 | 220 | 312 | 0.690 | 140 | 394 |

CN | 0.125 | 248 | 473 | 0.093 | 137 | 268 |

CO | 0.081 | 404 | 656 | 0.375 | 89 | 375 |

Code | Name of Coverage | Description |
---|---|---|

BC | Building and Contents | This coverage provides insurance for buildings and the properties within. In case the policyholder has purchased a rider, claims in this group may reflect additional amounts covered under endorsements. |

IM | Contractor’s Equipment | IM, an abbreviation for “inland marine” is used as the coverage code for equipments coverage, which originally belong to contractors. |

C | Collision | This provides coverage for impact of a vehicle with an object, impact of vehicle with an attached vehicle, or overturn of a vehicle. |

P | Comprehensive | Direct and accidental loss or damage to motor vehicle, including breakage of glass, loss caused by missiles, falling objects, fire, theft, explosion, earthquake, windstorm, hail, water, flood, malicious mischief or vandalism, riot or civil common, or colliding with a bird or animal. |

N | New | This code is used as an indication that the coverage is for vehicles of current model year, or 1∼2 years prior to the current model year. |

O | Old | This code is used as an indication that the coverage is for vehicles three or more years prior to the current model year. |

BC | IM | PN | PO | CN | |
---|---|---|---|---|---|

IM | 0.506 | ||||

PN | 0.465 | 0.584 | |||

PO | 0.490 | 0.590 | 0.771 | ||

CN | 0.492 | 0.541 | 0.679 | 0.566 | |

CO | 0.559 | 0.601 | 0.642 | 0.668 | 0.646 |

BC Frequency | IM Frequency | PN Frequency | PO Frequency | CN Frequency | CO Frequency | |
---|---|---|---|---|---|---|

BC Severity | −0.033 | 0.029 | −0.063 | −0.069 | 0.020 | −0.050 |

IM Severity | −0.033 | −0.078 | 0.110 | 0.249 | 0.159 | 0.225 |

PN Severity | 0.074 | 0.275 | −0.146 | −0.216 | 0.119 | 0.143 |

PO Severity | 0.111 | 0.171 | −0.161 | −0.119 | 0.258 | 0.137 |

CN Severity | −0.112 | −0.174 | −0.003 | 0.135 | 0.032 | −0.175 |

CO Severity | −0.099 | −0.079 | −0.055 | −0.083 | −0.068 | −0.032 |

BC | IM | PN | PO | CN | |
---|---|---|---|---|---|

IM | 0.220 | ||||

PN | 0.098 | 0.095 | |||

PO | 0.229 | 0.118 | 0.415 | ||

CN | 0.084 | 0.237 | 0.166 | 0.200 | |

CO | 0.132 | 0.261 | 0.075 | 0.140 | 0.244 |

BC | IM | PN | PO | CN | CO | |
---|---|---|---|---|---|---|

Coverage > 0 | 5660 | 4622 | 1638 | 2138 | 1529 | 2013 |

Average Severity > 0 | 1684 | 236 | 315 | 263 | 370 | 362 |

Variable Name | Description | Mean |
---|---|---|

lnCoverageBC | Log of the building and content coverage amount. | $37.050$ |

lnCoverageIM | Log of the contractor’s equipment coverage amount. | $0.848$ |

lnCoveragePN | Log of the comprehensive coverage amount for new vehicles. | $0.158$ |

lnCoveragePO | Log of the comprehensive coverage amount for old vehicles. | $0.614$ |

lnCoverageCN | Log of the collision coverage amount for new vehicles. | $0.096$ |

lnCoverageCO | Log of the collision coverage amount for old vehicles. | $0.305$ |

NoClaimCreditBC | Indicator for no building and content claims in prior year. | $0.328$ |

NoClaimCreditIM | Indicator for no contractor’s equipment claims in prior year. | $0.421$ |

NoClaimCreditPN | Indicator for no comprehensive claims for new cars in prior year. | $0.110$ |

NoClaimCreditPO | Indicator for no comprehensive claims for old cars in prior year. | $0.170$ |

NoClaimCreditCN | Indicator for no collision claims for new cars in prior year. | $0.090$ |

NoClaimCreditCO | Indicator for no collision claims for old cars in prior year. | $0.140$ |

EntityType | City, County, Misc, School, Town (Categorical) | |

lnDeductBC | Log of the BC deductible level, chosen by the entity. | $7.137$ |

lnDeductIM | Log of the IM deductible level, chosen by the entity. | $5.340$ |

**Table 9.**Comparison between Empirical Values and Expected Values for the building and contents (BC) Line.

Empical | ZeroinflPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB | |
---|---|---|---|---|---|---|---|

0 | 3976 | $4038.125$ | $3975.403$ | $3709.985$ | $4075.368$ | $4093.699$ | $3996.906$ |

1 | 997 | $754.384$ | $1024.219$ | $1012.267$ | $809.077$ | $791.424$ | $1003.169$ |

2 | 333 | $355.925$ | $276.082$ | $417.334$ | $313.359$ | $314.618$ | $280.600$ |

3 | 136 | $187.897$ | $146.962$ | $202.288$ | $155.741$ | $157.282$ | $136.758$ |

4 | 76 | $106.780$ | $82.052$ | $106.874$ | $88.866$ | $89.615$ | $75.822$ |

5 | 31 | $63.841$ | $48.426$ | $60.160$ | $55.484$ | $55.697$ | $46.021$ |

6 | 19 | $39.850$ | $30.212$ | $36.540$ | $36.919$ | $36.845$ | $29.854$ |

7 | 19 | $26.082$ | $19.850$ | $24.261$ | $25.765$ | $25.553$ | $20.379$ |

8 | 16 | $18.025$ | $13.670$ | $17.440$ | $18.663$ | $18.395$ | $14.482$ |

9 | 5 | $13.165$ | $9.808$ | $13.222$ | $13.932$ | $13.652$ | $10.632$ |

10 | 7 | $10.087$ | $7.269$ | $10.305$ | $10.664$ | $10.393$ | $8.016$ |

11 | 2 | $8.007$ | $5.505$ | $8.124$ | $8.336$ | $8.084$ | $6.180$ |

12 | 4 | $6.505$ | $4.219$ | $6.427$ | $6.636$ | $6.406$ | $4.855$ |

13 | 5 | $5.357$ | $3.248$ | $5.086$ | $5.367$ | $5.159$ | $3.875$ |

14 | 5 | $4.441$ | $2.502$ | $4.024$ | $4.401$ | $4.214$ | $3.136$ |

15 | 2 | $3.690$ | $1.925$ | $3.182$ | $3.653$ | $3.485$ | $2.569$ |

16 | 4 | $3.062$ | $1.479$ | $2.519$ | $3.066$ | $2.914$ | $2.127$ |

17 | 3 | $2.530$ | $1.134$ | $1.999$ | $2.598$ | $2.460$ | $1.777$ |

18 | 1 | $2.077$ | $0.867$ | $1.597$ | $2.221$ | $2.095$ | $1.498$ |

$\ge 19$ | 19 | $10.168$ | $5.167$ | $16.366$ | $19.876$ | $18.004$ | $11.343$ |

0 proportion | $0.702$ | $0.713$ | $0.702$ | $0.655$ | $0.720$ | $0.723$ | $0.706$ |

1 proportion | $0.176$ | $0.133$ | $0.181$ | $0.179$ | $0.143$ | $0.140$ | $0.177$ |

ZeroinfPoisson | ZeroonePoisson | Poisson | NB | ZeroinflNB | ZerooneNB |
---|---|---|---|---|---|

154.573 | 77.064 | 105.201 | 88.086 | 98.400 | 34.515 |

Variable Name | Coef. | Standard | ||
---|---|---|---|---|

Error | ||||

GB2 | (Intercept) | 5.620 | 0.199 | *** |

lnCoverageBC | 0.136 | 0.029 | *** | |

NoClaimCreditBC | 0.143 | 0.076 | . | |

lnDeductBC | 0.321 | 0.034 | *** | |

EntityType: City | −0.121 | 0.090 | ||

EntityType: County | −.059 | 0.112 | ||

EntityType: Misc | 0.052 | 0.142 | ||

EntityType: School | 0.182 | 0.092 | * | |

EntityType: Town | −0.206 | 0.141 | ||

σ | 0.343 | 0.070 | ||

${\alpha}_{1}$ | 0.486 | 0.119 | ||

${\alpha}_{2}$ | 0.349 | 0.083 | ||

NB | (Intercept) | −0.798 | 0.198 | *** |

lnCoverageBC | 0.853 | 0.033 | *** | |

NoClaimCreditBC | −0.400 | 0.132 | ** | |

lnDeductBC | −0.232 | 0.035 | *** | |

EntityType: City | −0.074 | 0.090 | ||

EntityType: County | 0.015 | 0.117 | ||

EntityType: Misc | −0.513 | 0.188 | ** | |

EntityType: School | −1.056 | 0.094 | *** | |

EntityType: Town | −0.016 | 0.160 | ||

log(size) | 0.370 | 0.115 | ||

Zero | (Intercept) | −6.928 | 0.840 | *** |

CoverageBC | −0.408 | 0.135 | ** | |

lnDeductBC | 0.880 | 0.108 | *** | |

NoClaimCreditBC | 0.954 | 0.459 | * | |

One | (Intercept) | −5.466 | 0.965 | *** |

CoverageBC | 0.142 | 0.117 | ||

lnDeductBC | 0.323 | 0.137 | * | |

NoClaimCreditBC | 0.669 | 0.447 |

Copula 1 | Copula 2 | $95\%$ Interval | |
---|---|---|---|

Gaussian | t(df = 3) | −0.0307 | −0.0122 |

Gaussian | t(df = 4) | −0.0202 | −0.0065 |

Gaussian | t(df = 5) | −0.0147 | −0.0038 |

Gaussian | t(df = 6) | −0.0114 | −0.0023 |

Variable Name | Coef. | Standard | ||
---|---|---|---|---|

Error | ||||

GB2 | (Intercept) | 5.629 | 0.195 | *** |

lnCoverageBC | 0.144 | 0.029 | *** | |

NoClaimCreditBC | 0.222 | 0.076 | ** | |

lnDeductBC | 0.320 | 0.031 | *** | |

EntityType: City | −0.148 | 0.090 | . | |

EntityType: County | −0.043 | 0.111 | ||

EntityType: Misc | 0.158 | 0.143 | ||

EntityType: School | 0.225 | 0.092 | * | |

EntityType: Town | −0.218 | 0.141 | ||

σ | 0.343 | 0.070 | ||

${\alpha}_{1}$ | 0.486 | 0.119 | ||

${\alpha}_{2}$ | 0.349 | 0.083 | ||

NB | (Intercept) | −0.789 | 0.083 | *** |

lnCoverageBC | 1.003 | 0.001 | *** | |

NoClaimCreditBC | −0.297 | 0.172 | . | |

lnDeductBC | −0.230 | 0.001 | *** | |

EntityType: City | −0.068 | 0.097 | ||

EntityType: County | −0.489 | 0.109 | *** | |

EntityType: Misc | −0.468 | 0.202 | * | |

EntityType: School | −0.645 | 0.083 | *** | |

EntityType: Town | 0.267 | 0.166 | ||

log(Size) | 0.370 | 0.115 | ||

Zero | (Intercept) | −6.246 | 0.364 | *** |

lnCoverageBC | −0.338 | 0.047 | *** | |

lnDeductBC | 0.910 | 0.050 | *** | |

NoClaimCreditBC | 0.888 | 0.355 | * | |

One | (Intercept) | −5.361 | 0.022 | *** |

lnCoverageBC | 0.345 | 0.013 | *** | |

lnDeductBC | 0.335 | 0.010 | *** | |

NoClaimCreditBC | 0.556 | 0.431 | ||

ρ | Dependence | −0.132 | 0.033 | *** |

IM | PN | PO | CN | CO | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Coef. | Std. Error | Coef. | Std. Error | Coef. | Std. Error | Coef. | Std. Error | Coef. | Std. Error | |||||||

GB2 | (Intercept) | 8.153 | 0.823 | *** | 7.918 | 0.046 | *** | 7.554 | 0.092 | *** | 6.773 | 0.059 | *** | 9.334 | 0.000 | *** |

lnCoverage | 0.304 | 0.065 | *** | 0.078 | 0.045 | . | 0.081 | 0.057 | 0.137 | 0.039 | *** | 0.161 | 0.000 | *** | ||

NoClaimCredit | 0.190 | 0.202 | 0.021 | 0.209 | 0.695 | 0.194 | *** | 0.140 | 0.144 | −0.296 | 0.001 | *** | ||||

lnDeduct | 0.028 | 0.125 | ||||||||||||||

σ | 0.955 | 0.365 | 0.047 | 0.043 | 0.100 | 0.130 | 0.863 | 0.513 | 40.193 | 31.080 | ||||||

${\alpha}_{1}$ | 1.171 | 0.630 | 0.054 | 0.050 | 0.102 | 0.137 | 4.932 | 6.441 | 0.038 | 0.030 | ||||||

${\alpha}_{2}$ | 1.337 | 0.856 | 0.076 | 0.068 | 0.108 | 0.145 | 1.279 | 1.131 | 0.025 | 0.019 | ||||||

NB | (Intercept) | −1.331 | 0.594 | * | −2.160 | 0.284 | *** | −2.664 | 0.297 | *** | −0.467 | 0.158 | ** | −1.746 | 0.187 | *** |

Coverage | 0.796 | 0.077 | *** | 0.239 | 0.065 | *** | 0.490 | 0.067 | *** | 0.487 | 0.054 | *** | 0.782 | 0.056 | *** | |

NoClaimCredit | −0.371 | 0.141 | ** | −0.588 | 0.194 | ** | −0.612 | 0.177 | *** | −0.668 | 0.157 | *** | −0.324 | 0.139 | * | |

lnDeduct | −0.140 | 0.085 | . | |||||||||||||

EntityType: City | −0.306 | 0.235 | 0.574 | 0.330 | . | 0.411 | 0.376 | 0.433 | 0.186 | * | 0.680 | 0.232 | ** | |||

EntityType: County | 0.139 | 0.274 | 3.083 | 0.294 | *** | 2.477 | 0.329 | *** | 1.131 | 0.172 | *** | 1.284 | 0.211 | *** | ||

EntityType: Misc | −2.195 | 1.024 | * | −0.060 | 0.642 | −0.508 | 0.709 | −0.323 | 0.456 | 0.486 | 0.442 | |||||

EntityType: School | −0.032 | 0.292 | 0.389 | 0.297 | 0.926 | 0.327 | ** | −0.192 | 0.185 | 1.350 | 0.208 | *** | ||||

EntityType: Town | −0.405 | 0.277 | −0.579 | 0.481 | −1.022 | 0.650 | −1.529 | 0.385 | *** | −0.450 | 0.355 | |||||

size | 0.724 | 1.004 | 0.766 | 1.420 | 1.302 | |||||||||||

ρ | Dependence | −0.109 | 0.097 | −0.154 | 0.064 | * | −0.166 | 0.073 | * | 0.171 | 0.064 | ** | 0.009 | 0.045 |

BC | IM | PN | PO | CN | |
---|---|---|---|---|---|

IM | $0.190$ | ||||

PN | $0.141$ | $0.162$ | |||

PO | $0.054$ | $0.206$ | $0.379$ | ||

CN | $0.101$ | $0.149$ | $0.271$ | $0.081$ | |

CO | $0.116$ | $0.213$ | $0.151$ | $0.231$ | $0.297$ |

BC | IM | PN | PO | CN | |
---|---|---|---|---|---|

IM | $0.145$ | ||||

PN | $0.134$ | $0.051$ | |||

PO | $0.298$ | $0.099$ | $0.498$ | ||

CN | $0.062$ | $0.110$ | $0.156$ | $0.168$ | |

CO | $0.106$ | $0.215$ | $0.083$ | $0.080$ | $0.210$ |

BC | IM | PN | PO | CN | |
---|---|---|---|---|---|

IM | $0.210$ | ||||

PN | $0.279$ | $0.367$ | |||

PO | $0.358$ | $0.412$ | $0.559$ | ||

CN | 0.265 | 0.266 | 0.553 | 0.328 | |

CO | $0.417$ | $0.359$ | $0.496$ | $0.562$ | $0.573$ |

BC | IM | PN | PO | CN | CO | Total | |
---|---|---|---|---|---|---|---|

Independent Tweedie | 0.410 | 0.304 | 0.602 | 0.461 | 0.512 | 0.482 | 0.500 |

Dependent Tweedie (Monte Carlo) | 0.412 | 0.305 | 0.601 | 0.462 | 0.511 | 0.481 | 0.501 |

Independent Frequency-Severity | 0.440 | 0.308 | 0.590 | 0.475 | 0.525 | 0.469 | 0.498 |

Dependent Frequency-Severity (Monte Carlo) | 0.435 | 0.308 | 0.590 | 0.477 | 0.525 | 0.485 | 0.521 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Frees, E.W.; Lee, G.; Yang, L.
Multivariate Frequency-Severity Regression Models in Insurance. *Risks* **2016**, *4*, 4.
https://doi.org/10.3390/risks4010004

**AMA Style**

Frees EW, Lee G, Yang L.
Multivariate Frequency-Severity Regression Models in Insurance. *Risks*. 2016; 4(1):4.
https://doi.org/10.3390/risks4010004

**Chicago/Turabian Style**

Frees, Edward W., Gee Lee, and Lu Yang.
2016. "Multivariate Frequency-Severity Regression Models in Insurance" *Risks* 4, no. 1: 4.
https://doi.org/10.3390/risks4010004