The Analysis of Development of Insurance Contract Premiums of General Liability Insurance in the Business Insurance Risk

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1 The Aalyss of Developmet of Isurace Cotract Premums of Geeral Lablty Isurace the Busess Isurace Rsk the Frame of the Czech Isurace Market Scetfc Coferece Jue, Pavla Kubová Departmet of Isurace Maagemet Techcal Uversty of Lberec, Faculty of Ecoomcs Lberec, Czech Republc pavla.kubova@tul.cz Abstract Ths paper deals wth the tme seres aalyss ad ther developmet predcto of surace cotract premum of geeral lablty surace the busess surace rsk the frame of the Czech surace market for years 01 ad 013. The tme seres are defed as a sequece of data pots, measured typcally at successve tmes, spaced at tme tervals. Data ths modelg are gross premum wrtte of lablty surace of employers for work jures ad occupatoal dseases of members of CAP (Czech Isurace Assocato) years 1998 to 011. Ths aalyss does ot clude ecoomc factors (for example: flato, ecoomc progress, ecoomc recesso, ecoomc shocks). Keywords- tme seres aalyss; predcto; cotract premum; geeral lablty surace, busess surace rsk. I. INTRODUCTION Whe characterzg the Czech surace market, several basc ecoomc dcators wll appear. For example, gross premum wrtte of surace cotract premum of geeral lablty surace the busess surace rsk. I ths paper, gross premum wrtte surace cotract premum of geeral lablty surace the busess surace rsk wth years 1998 to 011 wll be aalyzed ad ther developmet predcto for years 01 ad 013 wll be gve. The data for ths aalyss are used from the Czech Isurace Assocato (CAP). The aalyss s developed for the Studet Project Grat Competto 013; grat No Tme seres aalyss s dscussed may textbooks, see Hamlto (1994) [1]; Hdls, Hroová ad Novák (000) []; Chatfeld (003) [3] ad Tsay (005) [4]. I the frst part of ths paper, basc characterstc developmet of tme seres wll be aalyzed. The secod part wll by focused o detfcato of the tred by meas of hypotheses tests, tha a acceptable model wth predcto for years 01 ad 013 wll be chose. The estmate of tred fucto values wll be aalyzed by usg the statstc program Statgraphc Ceturo XVI. I the fal tables R.M.S.E. (root mea square error), I_adjusted^ (adjusted dex of determato), t-tests (tests crtero), P-values (crtcal sgfcace lmts) ad total F-test wll be calculated. Kara Mužáková Departmet of Isurace Maagemet Techcal Uversty of Lberec, Faculty of Ecoomcs Lberec, Czech Republc kara.muzakova@tul.cz II. TIME SERIES ANALYSIS For calculato of basc characterstc developmet of tme seres t s ecessary aalyze data about developmet of gross premum wrtte of surace cotract premum of geeral lablty surace the busess surace rsk years 1998 to 011 (see Tab. I). TABLE I. DEVELOPMENT OF GROSS PREMIUM WRITTEN OF BUSINESS INSURANCE RISK Gross premum wrtte of busess surace rsk ( thousads CZK) (y t) Source: CAP ( ) [5] Usg the vsual aalyss of the graphc record durg the tme seres ca recogze as a log-term tred durg the seres. You ca also motor some perodc developmetal chages. The followg Fg. 1 shows the progress of busess surace premums the years 1998 to

2 Gross premum wrtte of busess surace rsk ( thousads CZK) (yt) Scetfc Coferece Jue, Fgure 1. Developmet of Gross Premum Wrtte of Busess Isurace Rsk However, t must be sad that ths vsualzato s ever eough to kow the deeper coectos ad mechasms of the process. The elemetal characterstcs clude: dfferece of frst ad secod order, the rate of growth / decle, pace of ga / loss ad average, average rate of growth / decle, average absolute crease / decrease. I the text below, these characterstcs descrbed more detal Table are already calculated specfc values. The frst dfferece (1) characterzes the cremet value of the dcator tme seres for a certa perod wth the perod mmedately precedg. I other words, t tells us how uts of measure decreased or creased value. 1 t yt yt 1, t,3,...,. If the seres shows a certa developmetal tedeces, we ca derve from frst dffereces the secod or thrd dfferece. Accelerato s determed by comparg the absolute cremets, as the secod (absolute) dffereces. The secod dfferece () states the umber of uts decreased or creased value of the frst dfferece. 1 1 Δ t Δt Δ t 1 (yt y t 1 ) (yt 1 y t ), t 3, 4,...,. Growth coeffcet expressed percetage s called the coeffcet of growth (3). Idcates the percetage creased value of the tme seres at tme t from the prevous perod. y k t t, t, 3,...,. y t1 Other characterstcs are descrbed relatve addtos to the delght of growth ( T yt ) determg a rato betwee that ad the prevous member of the seres. These are percetages coeffcet growth. If the growth rate multpled by 100, dcates the percetage of the value at tme t 1, creased value at tme t. Growth rate (4) dcates the percetage creased or decreased value of the dcator. yt Tyt100 (4) As the aggregate characterstc of relatve chages for the etre tme seres of reports the average growth dex (5), whch s the geometrc average of the dvdual coeffcets of growth. 1 1 y k k1 k... k 1 (5) y1 The mea absolute crease (6) s the average aual crease or decrease value for the perod studed. All defed basc characterstcs are gve Tab. II. Year (t) 1 y y d d TABLE II. DEVELOPMENT OF ELEMENTAL CHARACTERISTICS OF THE GROSS PREMIUM WRITTEN OF BUSINESS INSURANCE RISK (y t) 1 t t , ,065 10,6531, , , , , , , , , , , , , , , , , , , , , , , ,3086 8, , ,3384 4, , , , , ,4957 3,4957 From the above Tab. II ca be see the largest crease surveyed values for the perod 1998 to ad 005 (compared to the prevous perod, the bggest crease beg 005, a crease of more tha 0.5 bllo CZK). The secod largest growth market surace busess surace was observed durg the reportg perod ( terms of premums) 004. Growth rate shows the percetage creased or decreased the value of the vestgated dcators. As already metoed, the hghest crease gross wrtte premums for geeral lablty surace (busess surace) for the reportg perod was recorded 005, the growth rate values examed dcators (prevous year) amouted to more tha 19 %. The rapd decle recorded surace market surace busess durg the reportg perod 006, whe the rate of decrease values researched dcators (prevous year) amouted to 3.83 %. The average growth rate, whch characterzes the average growth of the parameter, s The mea absolute crease for the perod s examed after roudg thousad CZK. kt Tyt (6) yt

3 Scetfc Coferece Jue, III. MODELING THE TREND OF THE TIME SERIES The tred detfcato was aalyzed by the program Statgraphcs Ceturo. The results of tests of dvdual tred fuctos parameters ca be fd Tab. III. TABLE III. LINEAR, QUADRATIC AND EXPONENTIAL TREND Tred Lear tred Quadratc tred Expoetal tred Tred fucto T t = a + bt T t = a + bt + ct (a + bt) T t = e t From the above Tab. III shows that the value of R.M.S.E. (7), the root mea square error (Root Mea Squared Error) s lowest for quadratc tred. Value R.M.S.E. s calculated accordg to the formula: R. M. S. E. t1 t t y T To test a sutable model was also used for determato dex (8). The hgher the dex value determato closer to the umber oe (or 100 %), the better the model captures the tred of the tme seres ad vce versa. R t t e (1,11 + 0,070457t) R.M.S.E R modf.(%) 95,663 95, ,109 H 0 a = 0 a = 0 a = 0 H 1 a 0 a 0 a 0 a ,11 T-test -16,8345 8, ,985 P-value 0,0000 < 0,05 0, < 0,05 0,0000 < 0,05 ( yˆ y) ( y y) H 0 b = 0 b = 0 b = 0 H 1 b 0 b 0 b 0 b , T-test 16,9617 5, ,4456 P-value 0,0000 < 0,05 0, < 0,05 0,0000 < 0,05 H 0 c = 0 c = 0 c = 0 H 1 c 0 c 0 c 0 T-test -1,185 P-value 0,85411 > 0,05 Dsapprove H 1, prove H 0. (7) (8) Lack of determato coeffcet (8) s that t depeds o the umber of model parameters (tred fucto). Ths defcecy removes the modfed dex determato (9) the form: 1 Rmod. 1 (1 R ) (9) p Determato dex s foud the rage: <0, 1>. The strogest depedece follow a lear model (R modfed value s hghest). I Tab. IV we are testg the hypotheses H 0 ad H 1. We are usg F-test to fd the sutablty of the model lear, quadratc or expoetal. H 0 TABLE IV. The lear acceptable model. TESTING A SUITABLE MODEL The quadratc acceptable model. The expoetal acceptable model. H 1 No H 0 No H 0 No H 0 F-test 87,70 147,61 08,67 P-value 0,0000 < 0,05 0,0000 < 0,05 0,0000 < 0,05 prove H 1. Accordg F-test, the ull hypothess s rejected. It s ecessary to proceed selectg a approprate model. Aother opto s a automatc model selecto the program Statgraphcs. Accordg to the method for selectg crtera such as Akake formato crtero (Akake, 1974) [6] s based o a lear model does ot stck. Akake formato crtero provdes formato about the relatve approprateess of the statstcal model, other words, represets the relatve rate of loss of formato descrbg realty usg the model. The geeral formula for calculatg the Akake formato crtero (10): A. I. C. k l( L) (10) where k s the umber of parameters of the statstcal model, ad L s the maxmum value of the lkelhood fucto for the estmated model. Akake formato crtero tells us that the compared statstcal models seems to be the best, but says othg about how ad whether a partcular model correspods to the observed data. I other words, f all the compared models descrbe the real data poorly, the value of the Akake formato crtero by us of ths fact does ot warg would oly be able to decde whch of these "bad" models correspodg to the data set relatve best. The followg Tab. V t ca be cosulted pot ad terval forecast for 01 ad 013 ad the lower ad upper cofdece lmt of 95 %. The Tab. VI shows pot ad terval forecast for 01 ad 013 ad the lower ad upper cofdece lmt of 99 %

4 Scetfc Coferece Jue, TABLE V. LINEAR TREND WITH FORECASTS FOR 01 AND 013 Low lmt, 95 % Upper lmt, 95 % TABLE VI. LINEAR TREND WITH FORECASTS FOR 01 AND 013 Low lmt, 99 % Upper lmt, 99 % The followg Fg. shows the lear tred wth two-year forecasts wth 95 % cofdece ). The above lear tred forecastg process eve for the 99% cofdece terval. Tab. VIII shows the forecast for 01 ad 013. TABLE VIII. QUADRATIC TREND WITH FORECASTS FOR 01 AND 013 WITH 99 % CONFIDENCE Low lmt, 99% Upper lmt, 99 % The followg Fg. 3 shows a lear tred wth a two-year forecasts. For 01, t s predcted gross surace premums vestgated 4,3 bllo CZK ad 013 CZK 4.5 bllo, wth a cofdece terval of 99%. It would be very terestg to compare whether ths predcto came true 01, but the data are ot yet avalable. Fgure. Lear Tred wth Two-year s The crtera that take to accout data (ulke A.I.C.) are: M.S.E. (mea squared error), M.A.E. (mea absolute error), M.A.P.E. (mea absolute percetage error). For automatc model selecto the program Statgraphcs we have chose the M.S.E. crtero. The average squared error (M.S.E.) of the estmate s oe of the ways to quatfy the dfferece betwee the values resultg by estmatg a true value of that estmate. M.S.E. evaluates the dameter squared errors. The lowest value M.S.E. accordg to the calculatos Tab. III s based o quadratc tred. The followg Tab. VII t s show predctos for 01 ad 013, just as the quadratc model. TABLE VII. QUADRATIC TREND WITH FORECASTS FOR 01 AND 013 WITH 95 % CONFIDENCE Year (t) Low lmt, 95% Upper lmt, 95 % The above values derved from the aalyzed treds (lear ad quadratc) s cled to the lear tred, maly because of the ull hypothess prove the quadratc tred of the parameter c (the value of P-value > 0.05, amely Fgure 3. Lear Tred wth Two-year s IV. CONCLUSION The ma am of ths paper was to aalyze the developmet of surace cotract premums of geeral surace lablty, amely busess surace premums, the perod from wth predcto of gross premums wrtte for the years 01 ad 013. I the frst part ot ths paper have bee detfed the basc characterstcs, amely the dfferece of the frst ad secod order rate of growth / decle, the rate of crease / decrease, the average rate of growth / decle ad average absolute crease / decrease. The largest absolute creases were recorded 004 ad 005 (compared to the prevous year, the bggest ga the year of 005 by more tha CZK 0.5 bllo). The secod largest growth market surace busess surace was observed 004. The hghest growth was recorded 005 (growth rate amouted to more tha 19 %). The rapd decle recorded surace market surace busess 006 (the rate of decrease amouted to 3.83 %). Average growth rate (characterzes the average growth observed dcators for the perod) s the perod aalyzed The mea

5 Scetfc Coferece Jue, absolute crease for the perod s examed after roudg thousad. CZK. The secod part of the artcle have bee focused o modelg tred. Researched tred fucto has bee chose as the basc characterstcs of the developmet of the examed values, as follows: lear tred, quadratc tred, expoetal tred. Usg Statgraphcs Ceturo XVI. were calculated requred parameter values a, b ad c Furthermore, the calculatos R.M.S.E. values ad a modfed dex determato. To select the approprate model requred for the aalyss of tme seres, t was ecessary to evaluate the amout of R.M.S.E., respectvely, M.S.E., the the dex of modfed accordg to the results of determato ad P-value for each parameter a rejecto or acceptace of hypotheses. The lowest error was publshed o the quadratc tred fucto, whle the modfed determato dex was hghest the lear model. The best model for the aalyss tme was chose lear tred ad the reasos that the P-value was publshed o quadratc tred testg parameter c s hgh ( ) ad the ull hypothess (c = 0) was therefore adopted. Accordg to the results pot forecast for the selected lear model was a cofdece terval of 99 % for 01 predcted gross premums wrtte surace busess after roudg of CZK ad the roudg of CZK For comparso, f the predcted values correspod to realty, due to data uavalablty, we'll kow utl the ed of 013, whch wll be publshed the Aual Report of the Czech Isurace Assocato. The program Statgraphcs Ceturo XVI. was also used by the auto model selecto crtera such as the selecto method I chose the Akake formato crtero (A.I.C.). Accordg to the crtera A.I.C. publshed by automatc model selecto as well as the best model lear model. ACKNOWLEDGMENT Ths paper was created wth the support of the Studet Grat Competto 013 (SGS 013); grat Nr /013: "The Comparso of Methods Solutos Evrometal Rsks the Global Isurace Market". REFERENCES [1] J. D. Hamlto, Tme Seres Aalyss. Edto 1. Prceto Uversty Press, Prceto, ISBN [] R. Hdls; S. Hroová,; I. Novák, Metody statstcké aalýzy pro ekoomy. Edto. Maagemet Press, Praha, 000. ISBN [3] C. Chatfeld, The aalyss of tme seres: a troducto. Edto 6. CRC Press, Lodo, 003. ISBN [4] S. R. Tsay, Aalyss of facal tme seres. Edto. Joh Wley ad Sos, Chcago, 005. ISBN [5] CAP ( ). Aual Reports , avalable form: [6] H. Akake, A ew look at the statstcal model detfcato. IEEE Trasactos o Automatc Cotrol,