Actuarial loss modeling with gamma distributions and dependence structure analysis

Actuarial Science: Modeling

The objective of this report is to investigate dependence structure between the home line of business and domestic motor line of business to determine the global amount of capital to hold for both lines of business. The paper uses the regression analysis and descriptive statistics to analyze the raw data collected for the losses on both lines of business. The results from the analysis reveal that there is little or no positive dependence structure between the two lines of business because the R. Square value from the regression analysis output is 0.29, which is closer to 0. Moreover, there are higher losses in the domestic motor line of business than the home line of business. There is also higher volatility in the domestic motor line of business than home line of business. Thus, the report recommends that home line of business should attract more global capital than the domestic motor line of business.

Introduction

The objective of research is to use actuarial modeling and fit loss data into a gamma distribution to determine whether the dependence structure between the home line of business and domestic motor line of business is to be taking into account when determining the global amount of capital to hold for both lines. The paper establishes whether there is a positive dependence between home and domestic motor lines of business.

Assumption

Correlation between different lines of business in the United States is critical in assessing the aggregate of portfolio risks. Modeling is carried out using an analysis-synthesis paradigm. (Benfield, 2009).The analysis is carried out using the data on the losses on the domestic line of business and motor line of business using the regression analysis and descriptive statistics.

The assumption used to carry out the analysis between home line of business and domestic motor line of business is to use a linear relationship model with +1 indicating that a perfect linear relationship between home line of business and domestic motor line of business line. On the other hand, -1 indicates perfect decreasing relationships. The closer the coefficient to +1, the stronger the linear relationships between the two line of business. The study uses R-Square to establish the statistical confidence.

Analysis of the Data

The study carries out the statistical analysis on the raw data on losses of home line of business and losses of the domestic motor line of business using the descriptive statistics. The descriptive statistics assists is summarizing the whole raw data in a manageable form. Using the descriptive statistics, the paper has been able to present the mean, standard deviation, median and variance of the whole data. The descriptive statistics also assists in comparing the home line of business with the domestic motor line of business and determine the global amount of capital to hold in those business lines.

Table 1: Descriptive Statistics

L1 (losses for Home line of business)

L2 ( losses for Domestic Motor line of business)

Mean

Standard Error

15.7346499

21.0997206

Median

2092.83193

Mode

#N/A

#N/A

Standard Deviation

Sample Variance

8895025.82

15995081.3

Kurstosis

5.6835354

5.99088783

Skewness

1.96918409

1.98859162

Range

31738.8805

42425.0881

Minimum

0.05521071

0.07189203

Maximum

31738.9357

42425.16

Sum

107843395

144406272

Count

35928

35928

Level of Statistical Confidence (95.0%)

30.8403853

41.3560846

The Mode is the most repeated value in the data. From the table 1, the mode is not available because there is no repeated value in the data of the home line of business and domestic motor line of business data set. (North Carolina State University, 2004). Thus, the paper uses the Mean value to compare the losses in the home line business to the domestic motor line of business. The Analysis reveals that the domestic motor line of business records higher losses than the home line of business. Typically, the Mean value for the losses of the domestic motor line of business is 4019.32 while the Mean value for the losses of home line of business is 3001.6. From the Mean value, the losses in the domestic motor line of business are 14.6% greater than the losses of the home line of business (See Table 2).

Table 2

L1 (losses for Home line of business)

L2 ( losses for Domestic Motor line of business)

Difference

% Difference

Mean

3001.65 (42.7% )

4019.32 (57.3%)

14.6%

Standard Deviation

2982.45 (42.71%)

3999.38 (57.28)

11.57%

The paper also uses the Standard Deviation for the analysis. The standard deviation measures the spread of the raw data to the mean. The Standard Deviation is the total square root of the Sample Variance. Using the Standard Deviation of the raw data, it is revealed that there is a higher volatility in the losses in the domestic motor line of business than the home line of business. Based on the data from descriptive statistics, the Standard Deviation for the loss of home line of business is lower than the Standard Deviation for the loss of domestic motor line of business. From the data in the table 2, the losses of the home line of business are more stable because of the lower Standard Deviation. Thus, the Mean and Standard Deviation values make home line of business to be more favorable to global capital than the motor line of business. The paper further uses the regression analysis to derive more accurate results from the data.

Regression Analysis

The regression analysis provides more accurate strategies to achieve the data analysis. The paper uses the parameter such as R-Squared value to measure the level of influence that independent variable has on dependent variable. The closer of R-Squared value to 1, the stronger the linear relationships the dependent and independent variables. However, if the R-Squared is close to 0, there are no relationships between the data.