The Analysis of Development of Insurance Contract
Premiums of General Liability Insurance
in the Business Insurance Risk
in the Frame of the Czech Insurance Market in 1998–2011
Pavla Kubová
Department of Insurance Management Technical University of Liberec, Faculty of Economics
Liberec, Czech Republic pavla.kubova@tul.cz
Karina Mužáková
Department of Insurance Management Technical University of Liberec, Faculty of Economics
Liberec, Czech Republic karina.muzakova@tul.cz
Abstract— This paper deals with the time series analysis and their development prediction of insurance contract premium of general liability insurance in the business insurance risk in the frame of the Czech insurance market for years 2012 and 2013. The time series are defined as a sequence of data points, measured typically at successive times, spaced at time intervals. Data in this modeling are gross premium written of liability insurance of employers for work injuries and occupational diseases of members of CAP (Czech Insurance Association) in years 1998 to 2011. This analysis does not include economic factors (for example: inflation, economic progress, economic recession, economic shocks).
Keywords- time series analysis; prediction; contract premium; general liability insurance, business insurance risk.
I. INTRODUCTION
When characterizing the Czech insurance market, several basic economic indicators will appear. For example, gross premium written of insurance contract premium of general liability insurance in the business insurance risk. In this paper, gross premium written insurance contract premium of general liability insurance in the business insurance risk within years 1998 to 2011 will be analyzed and their development prediction for years 2012 and 2013 will be given. The data for this analysis are used from the Czech Insurance Association (CAP). The analysis is developed for the Student Project Grant Competition 2013; grant No. 38010. Time series analysis is discussed in many textbooks, see Hamilton (1994) [1]; Hindls, Hronová and Novák (2000) [2]; Chatfield (2003) [3] and Tsay (2005) [4].
In the first part of this paper, basic characteristic development of time series will be analyzed. The second part will by focused on identification of the trend by means of hypotheses tests, than an acceptable model with prediction for years 2012 and 2013 will be chosen. The estimate of trend function values will be analyzed by using the statistic program Statgraphic Centurion XVI. In the final tables R.M.S.E. (root mean square error), I_adjusted^2 (adjusted index of determination), t-tests (tests criterion), P-values (critical significance limits) and total F-test will be calculated.
II. TIME SERIES ANALYSIS
For calculation of basic characteristic development of time series it is necessary analyze data about development of gross premium written of insurance contract premium of general liability insurance in the business insurance risk in years 1998 to 2011 (see in Tab. I).
TABLE I. DEVELOPMENT OF GROSS PREMIUM WRITTEN OF BUSINESS INSURANCE RISK Year (t) Gross premium written of business insurance risk (in thousands CZK) (yt) 1998 1 724 346 1999 1 858 111 2000 1 900 203 2001 2 073 818 2002 2 301 347 2003 2 503 152 2004 2 911 805 2005 3 470 271 2006 3 337 113 2007 3 329 308 2008 3 606 335 2009 3 762 793 2010 3 895 223 2011 4 021 801 Source: CAP (1999–2011) [5] Using the visual analysis of the graphic record during the time series can recognize as a long-term trend during the series. You can also monitor some periodic developmental changes. The following Fig. 1 shows the progress of business insurance premiums in the years 1998 to 2011.
Figure 1. Development of Gross Premium Written of Business Insurance Risk
However, it must be said that this visualization is never enough to know the deeper connections and mechanisms of the process.
The elemental characteristics include: difference of first and second order, the rate of growth / decline, pace of gain / loss and average, average rate of growth / decline, average absolute increase / decrease.
In the text below, these characteristics described in more detail in Table 2 are already calculated specific values.
The first difference (1) characterizes the increment value of the indicator time series for a certain period with the period immediately preceding. In other words, it tells us how units of measure decreased or increased value.
If the series shows a certain developmental tendencies, we can derive from first differences the second or third difference. Acceleration is determined by comparing the absolute increments, as the second (absolute) differences. The second difference (2) states the number of units decreased or increased value of the first difference.
Growth coefficient expressed in percentage is called the coefficient of growth (3). Indicates the percentage increased value of the time series at time t from the previous period.
Other characteristics are described relative additions to the delight of growth (
T
yt) determining a ratio between that and the previous member of the series. These are percentages coefficient growth. If the growth rate multiplied by 100, indicates the percentage of the value at time t − 1, increased value at time t. Growth rate (4) indicates the percentage100
t t
y Ty
(4)As the aggregate characteristic of relative changes for the entire time series of reports the average growth index (5), which is the geometric average of the individual coefficients of growth.
(5) The mean absolute increase (6) is the average annual increase or decrease in value for the period studied. All defined basic characteristics are given in Tab. II.
1 2 1 1 1
n n n i i i y y d d n n (6)TABLE II. DEVELOPMENT OF ELEMENTAL CHARACTERISTICS OF THE GROSS PREMIUM WRITTEN OF BUSINESS INSURANCE RISK
From the above Tab. II can be seen the largest increase surveyed values for the period 1998 to 2011 in 2004 and 2005 (compared to the previous period, the biggest increase being in 2005, an increase of more than 0.5 billion CZK). The second largest growth market in insurance business insurance was observed during the reporting period (in terms of premiums) in 2004. Growth rate shows the percentage increased or decreased the value of the investigated indicators. As already mentioned, the highest increase in gross written premiums for general liability insurance (business insurance) for the reporting period was recorded in 2005, the growth rate values examined indicators (previous year) amounted to more than 19 %. The rapid decline recorded insurance market insurance business during the reporting period in 2006, when the rate of decrease values researched indicators (previous year) amounted to 3.83 %.
The average growth rate, which characterizes the average growth of the parameter, is 1.06731. The mean absolute increase for the period 1998−2011 is examined after rounding 176 727.3 thousand CZK. 1 500 000 2 000 000 2 500 000 3 000 000 3 500 000 4 000 000 4 500 000 1997 2000 2003 2006 2009 G ro ss p re m iu m w ritten of b u sin ess in su ra n ce r isk (in th o u sa n d s CZ K ) (yt ) Year (t) Year (t) (yt) 1998 1 724 346 × × × × × 1999 1 858 111 133 765 × × × 7,75743 2000 1 900 203 42 092 –91 673 1,02265 102,26531 2,26531 2001 2 073 818 173 615 131 523 1,09136 109,13666 9,13665 2002 2 301 347 227 529 53 914 1,10971 110,97150 10,97150 2003 2 503 152 201 805 –25 724 1,08769 108,76899 8,76899 2004 2 911 805 408 653 206 848 1,16325 116,32554 16,32554 2005 3 470 271 558 466 149 813 1,19179 119,17937 19,17937 2006 3 337 113 –133 158 –691 624 0,96162 96,162893 –3,83711 2007 3 329 308 –7 805 125 353 0,99766 99,766115 –0,23388 2008 3 606 335 277 027 284 832 1,08320 108,32086 8,32085 2009 3 762 793 156 458 –120 569 1,04338 104,33842 4,33842 2010 3 895 223 132 430 –24 028 1,03519 103,51946 3,51946 2011 4 021 801 126 578 –5 852 1,03249 103,24957 3,24957 1 1, 2, 3,..., . t yt yt t n 1 2 3 , . t t t y k , t , ,... n y 2 1 1 1 2 1 1 3 4 . t t t t t t t Δ Δ Δ (y y ) (y y ), t , ,...,n 1 1 1 1 2 1
n
n yn n yk
k k
... k
1 t 2 t k
tT
yt
ytIII. MODELING THE TREND OF THE TIME SERIES The trend identification was analyzed by the program Statgraphics Centurion. The results of tests of individual trend functions parameters can be find in Tab. III.
TABLE III. LINEAR, QUADRATIC AND EXPONENTIAL TREND
From the above Tab. III shows that the value of R.M.S.E. (7), the root mean square error (Root Mean Squared Error) is lowest for quadratic trend. Value R.M.S.E. is calculated according to the formula:
2 1 . . . .
n t t t y T R M S E n (7)To test a suitable model was also used for determination index (8). The higher the index value determination closer to the number one (or 100 %), the better the model captures the trend of the time series and vice versa.
2 2 1 2 1 ˆ ( ) ( )
n i i n i i y y R y y (8)Lack of determination coefficient (8) is that it depends on the number of model parameters (trend function). This deficiency removes the modified index determination (9) in the form: mod.2 1 (1 2) 1 n R R n p (9)
Determination index is found in the range: <0, 1>. The strongest dependence follow a linear model (R2 modified value is highest). In Tab. IV we are testing the hypotheses H0 and H1. We are using F-test to find the suitability of the model linear, quadratic or exponential.
TABLE IV. TESTING A SUITABLE MODEL
H0 The linear trend is not acceptable model. The quadratic trend is not acceptable model. The exponential trend is not acceptable model.
H1 Non H0 Non H0 Non H0
F-test 287,70 147,61 208,67 P-value 0,0000 < 0,05 0,0000 < 0,05 0,0000 < 0,05 Test conclusion Disapprove H0, prove H1. Disapprove H0, prove H1. Disapprove H0, prove H1.
According F-test, the null hypothesis is rejected. It is necessary to proceed in selecting an appropriate model. Another option is an automatic model selection in the program Statgraphics. According to the method for selecting criteria such as Akaike information criterion (Akaike, 1974) [6] is based on a linear model does not stick.
Akaike information criterion provides information about the relative appropriateness of the statistical model, in other words, represents the relative rate of loss of information in describing reality using the model. The general formula for calculating the Akaike information criterion (10):
A I C. . .2k2 ln( )L (10) where k is the number of parameters of the statistical model, and L is the maximum value of the likelihood function for the estimated model.
Akaike information criterion tells us that the compared statistical models seems to be the best, but says nothing about how and whether a particular model corresponds to the observed data. In other words, if all the compared models describe the real data poorly, the value of the Akaike information criterion by us of this fact does not warning would only be able to decide which of these "bad" models corresponding to the data set relative best.
The following Tab. V it can be consulted point and interval forecast for 2012 and 2013 and the lower and upper confidence limit of 95 %.
The Tab. VI shows point and interval forecast for 2012 and 2013 and the lower and upper confidence limit of 99 %. Trend Linear trend Quadratic trend Exponential trend
Trend function Tt = a + bt Tt = a + bt + ct2 Tt = e(a + bt) Forecast 1 456 840 000 + 193 332 000t 1 315 280 000 + 246 418 000t – 3 539 010t2 e (21,2211 + 0,070457t) R.M.S.E. 171 919 000 170 081 000 237 493 000 R2 modif.(%) 95,6623 95,7546 94,109 H0 a = 0 a = 0 a = 0 H1 a ≠ 0 a ≠ 0 a ≠ 0 a 1 456 840 000 1 315 280 000 21,2211 T-test -16,8345 8,29981 510,985 P-value 0,0000 < 0,05 0,000005 < 0,05 0,0000 < 0,05 Test
conclusion Disapprove Hprove H 0, 1. Disapprove H0, prove H1. Disapprove H0, prove H1. H0 b = 0 b = 0 b = 0 H1 b ≠ 0 b ≠ 0 b ≠ 0 b 193 332 000 246 418 000 0,070457 T-test 16,9617 5,06997 14,4456 P-value 0,0000 < 0,05 0,000361 < 0,05 0,0000 < 0,05 Test conclusion Disapprove H0, prove H1. Disapprove H0, prove H1. Disapprove H0, prove H1. H0 c = 0 c = 0 c = 0 H1 c ≠ 0 c ≠ 0 c ≠ 0 T-test -1,12285 P-value 0,285411 > 0,05 Test conclusion Disapprove H1, prove H0.
TABLE V. LINEAR TREND WITH FORECASTS FOR 2012 AND 2013 Year (t) Forecast (CZK) Low limit, 95 %
(CZK)
Upper limit, 95 % (CZK) 2012 4 356 820 000 3 926 680 000 4 786 970 000 2013 4 550 160 000 4 108 690 000 4 991 620 000 TABLE VI. LINEAR TREND WITH FORECASTS FOR 2012 AND 2013
Year (t) Forecast (CZK) Low limit, 99 % (CZK)
Upper limit, 99 % (CZK) 2012 4 356 820 000 3 753 790 000 4 959 860 000 2013 4 550 160 000 3 931 250 000 5 169 060 000 The following Fig. 2 shows the linear trend with two-year forecasts with 95 % confidence.
Figure 2. Linear Trend with Two-year Forecasts
The criteria that take into account data (unlike A.I.C.) are:
M.S.E. (mean squared error), M.A.E. (mean absolute error), M.A.P.E. (mean absolute percentage error). For automatic
model selection in the program Statgraphics we have chosen the M.S.E. criterion. The average squared error (M.S.E.) of the estimate is one of the ways to quantify the difference between the values resulting by estimating a true value of that estimate. M.S.E. evaluates the diameter squared errors.
The lowest value M.S.E. according to the calculations in Tab. III is based on quadratic trend. The following Tab. VII it is shown predictions for 2012 and 2013, just as the quadratic model.
TABLE VII. QUADRATIC TREND WITH FORECASTS FOR 2012 AND 2013 WITH 95 % CONFIDENCE
The above values derived from the analyzed trends (linear and quadratic) is inclined to the linear trend, mainly because of the null hypothesis prove the quadratic trend of the parameter c (the value of P-value > 0.05, namely
0.285411). The above linear trend forecasting process even for the 99% confidence interval. Tab. VIII shows the forecast for 2012 and 2013.
TABLE VIII. QUADRATIC TREND WITH FORECASTS FOR 2012 AND 2013 WITH 99 % CONFIDENCE Year (t) Forecast (CZK) Low limit, 99% (CZK) Upper limit, 99 % (CZK) 2012 4 356 820 000 3 753 790 000 4 959 860 000 2013 4 550 160 000 3 931 250 000 5 169 060 000 The following Fig. 3 shows a linear trend with a two-year forecasts. For 2012, it is predicted gross insurance premiums investigated 4,3 billion CZK and 2013 CZK 4.5 billion, with a confidence interval of 99%. It would be very interesting to compare whether this prediction came true in 2012, but the data are not yet available.
Figure 3. Linear Trend with Two-year Forecasts IV. CONCLUSION
The main aim of this paper was to analyze the development of insurance contract premiums of general insurance liability, namely business insurance premiums, in the period from 1998–2011with prediction of gross premiums written for the years 2012 and 2013.
In the first part ot this paper have been idenitfied the basic characteristics, namely the difference of the first and second order rate of growth / decline, the rate of increase / decrease, the average rate of growth / decline and average absolute increase / decrease. The largest absolute increases were recorded in 2004 and 2005 (compared to the previous year, the biggest gain in the year of 2005 by more than CZK 0.5 billion). The second largest growth market in insurance business insurance was observed in 2004. The highest growth was recorded in 2005 (growth rate amounted to more than 19 %). The rapid decline recorded insurance market insurance business in 2006 (the rate of decrease amounted to 3.83 %). Average growth rate (characterizes the average growth observed indicators for the period) is the period analyzed 1.06731. The mean Year (t) Forecast (CZK) Low limit, 95% (CZK) Upper limit, 95 % (CZK) 2012 4 215 260 000 3 703 610 000 4 726 920 000 2013 4 351 970 000 3 764 120 000 4 939 820 000
absolute increase for the period 1998–2011 is examined after rounding 176 727.3 thousand. CZK.
The second part of the article have been focused on modeling trend. Researched trend function has been chosen as the basic characteristics of the development of the examined values, as follows: linear trend, quadratic trend, exponential trend. Using Statgraphics Centurion XVI. were calculated required parameter values a, b and c Furthermore, the calculations R.M.S.E. values and a modified index determination. To select the appropriate model required for the analysis of time series, it was necessary to evaluate the amount of R.M.S.E., respectively, M.S.E., then the index of modified according to the results of determination and P-value for each parameter a rejection or acceptance of hypotheses. The lowest error was published on the quadratic trend function, while the modified determination index was highest in the linear model. The best model for the analysis time was chosen linear trend and the reasons that the P-value was published on quadratic trend testing parameter c is high (0.285411) and the null hypothesis (c = 0) was therefore adopted.
According to the results point forecast for the selected linear model was a confidence interval of 99 % for 2012 predicted gross premiums written insurance business after rounding of CZK 4 356 820 000 in 2013 and the rounding of CZK 4 550 160 000. For comparison, if the predicted values correspond to reality, due to data unavailability, we'll
know until the end of 2013, which will be published in the Annual Report of the Czech Insurance Association.
The program Statgraphics Centurion XVI. was also used by the auto model selection criteria such as the selection method I chose the Akaike information criterion (A.I.C.). According to the criteria A.I.C. published by automatic model selection as well as the best model linear model.
ACKNOWLEDGMENT
This paper was created with the support of the Student Grant Competition 2013 (SGS 2013); grant Nr. 38010/2013: "The Comparison of Methods Solutions Environmental Risks in the Global Insurance Market".
REFERENCES
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[5] CAP (1999–2011). Annual Reports 1999–2011, available form: www.cap.cz.
[6] H. Akaike, A new look at the statistical model identification. IEEE Transactions on Automatic Control, 1974.
