Predicting Number of Purchasing Life Insurance. Using Markov Chain Method

Transcription

1 Applied Matheatical Sciences, Vol. 8, 04, no. 8, HIKARI Ltd, Predicting Nber of Prchasing Life Insrance Using Markov Chain Method Mohd Rahiie Bin Md Noor and Zaidi at Isa Matheatical Departent, University Teknologi Mara Machang Kelantan; Acarial Sceinces Departent, University Kebangsaan Malaysia Bangi Selangor Corresponding athor Copyright 04 Mohd Rahiie Bin Md Noor and Zaidi at Isa. This is an open access article distribted nder the Creative Coons Attribtion License, which perits nrestricted se, distribtion, and reprodction in any edi, provided the original work is properly cited. Abstract This sdy describes the Markov chain approach applied in forecasting the life insrance bying patterns. This odel sing a saple prchase life insrance fro General Assrance Berhad covering the years Markov chain odel bilt is kind of the first stage with a hoogeneos tie. This odel ses the idea (stop-otion) to clarify the circstances of the nber and tie of prchase. At the end of the sdy show how the Markov chain to be a good ethod in predicting. Therefore, this ethod shold be extended in varios fields. Keywords: Markov-Chain; Stochastic Process; Hoogeneos Tie INTRODUCTION A copany of corse has the intention and goals to be achieved in the fre. They will ake every effort to increase profits fro year to year. However, a copany ay not be able to aintain a balance of profits earned stas withot providing an optiistic strategy and plan. Norally, analysts se cash flow data to predict fre profits earned by the copany. However, there is a concept that shold be addressed in the forecasting techniqes of cstoer behavior and cstoer anageent. Forecasting techniqe is said to be synonyos with bsiness becase

2 4088 Mohd Rahiie Bin Md Noor and Zaidi at Isa they can help bsiness copetition strategy for gaining the advantage. The robstness strategy and strong financial resorces can help attract ore cstoers sing the copany's prodcts. This condition st be aintained in order to axiize profits and pt the copany on solid perforance clsters. The copany is also able to predict the pattern of cstoer prchases of the prodct arket. Manageent strategy to attract cstoers bilt taking into accont cstoer behavior (Cstoer Life Vale, CLV ) as well as providing a satisfactory cstoer service. CLV concept is inflenced by the anageent of the cstoer (Cstoer Relationship anageent, CRM ). Dwyer (989 ) in writing stating cstoer deposits will inflence cstoer bying patterns. High savings rate that akes a person so obsessed to increase the nber of prchases above reqireents shold. Conseqently, a odel was constrcted to predict the nber of cstoers that reain with the insrance copany and get insrance coverage. Constrction of this odel sing the estiated probability of the cstoer to ake prchases in the fre. Cstoers will prchase the policy and receive the protection of all contracal agreeents with insrance copanies. Therefore, this sdy will se the odel to find patterns insrance prchase behavior of cstoers. In addition, the adaptive Markov chain odel can be seen fro the first stage of basic stationary Markov chain which will be explained in the literare review chapter. It also describes the tests to be done to validate the odel bilt sing the appropriate tie period. Dration and interval shold be set in the best possible becase it affects the accracy of the prediction. Therefore, this odel is sed to predict the dynaics of the insrance prchase ade by a cstoer sing the transition probability atrix. The statistical test sed to look for sitable odel with the data collected is iobilization test or hoogeneos. Insrance Prior sdies have been ade, we need to know and nderstand the fndaentals of insrance. Insrance is the transfer of risk by an individal or an organization known as the policyholder to the insrance copany. In rern, the insrance copany receives payent in the for of prei. If the policyholders bear the loss, the insrance copany will pay copensation for loss or daage. There are two types of insrance, naely life and general insrance (Takafl Association, 003). The srvey data sed in this sdy is taken fro the General Assrance Berhad. These data inclde the total nber of cstoers in 003, 004, 005 and 006 for life insrance. i. Methodology Stochastic process is a collection of rando variables, where t indexed in the set T (set paraeters or set the tie). The vales taken by the so-called process conditions and for the state space (all conditions) for the process (AK Bas, 003). Stochastic theory and the theory of probability can be sed to describe the evoltion of the odel syste over tie. A discrete tie stochastic process acally describes the relationship between pebolebah (Winston, 994). However, not known for certain ahead of tie and is known for rando variables. Discrete tie stochastic process for explaining the relationship rando variables. (Winston, 3rd). The relations of these variables is a condition that occrs in tie (0,,,...)

3 Predicting nber of prchasing life insrance 4089 Figre : Discrete Stochastic Processes Happens In Tie, t, t + 0 t t + X X t X t Markov chain is a stochastic process that occrs in discrete tie. Stochastic syste in the Markov chain is said to be dynaic when the probability of achieving a particlar siation depends on the previos period. In general, the stochastic process X t U for discrete tie. Markov chains t, t,...; U S,,...E can be expressed as follows : 0 in the discrete tie P[ X j X ( t0) s0, X ( t) s,..., X ( ) i it,..., X ii, X 0 i0 ] P( X j X ( t ) i) q (, ) = with and,then is the transition probability atrix. This is called the first stage of a stochastic process. In principle, the probability distribtion of the Markov chain depends on the circstances at the tie the state has passed p the chain to the siation at the tie. So, for a new stateent reslts fro the previos stateent. It is also agreed by Linios N. And G. Oprison (00) stating the expected fre state probabilities can be calclated by taking into accont the siation that has been traveled. Markov chain is said to be tie hoogeneos or fixed (stationary) if it has a constant period of tie. So, the probability of each of the processes occrring at the sae tie interval. q t, t q t, t, i j, Anderson and Goodan (957) describes the se of axi likelihood probability atrix in Markov chain data. Let n s 0, s..., s k is nbers of (ties), correlation eqations for each tie period are shown below : X t s0, X t s,..., X t k sk If we defined n s,..., s n s s and N s,..., s n s s 0 k 0,..., 0T k k 0 k 0,..., 0T k s, s..., s k as the nber of tie periods. While a seqence 0 the distribtion of the events that occrred at any tie in the saple data. N is the nber of tie periods and the expected axi likelihood of the transition atrix is : n i, j qˆ t, n ( i), i, j s, 0 T k

4 4090 Mohd Rahiie Bin Md Noor and Zaidi at Isa If the Markov chain has a fixed tie (stationary), the axi probability is: ni, j qˆ N( i), i, j s Nare of the Markov chain assption of fre state obtained fro the crrent state and the last state of a process. In addition, the Markov chain iobilization test was perfored to obtain the appropriate tie interval. Anderson and Goodan (957) have sed the chi-sqare test as a coparative hypothesis. The process is said to be stationary if the probability of a seqence of transitions occr in the sae tie period. Stationarity calclation sing chi-sqare test ( ) : ( ) E E j t n n i, j t, ni Ni n i, j n i Ni Transition atrix for each Markov chain has a row E and coln E. Degrees of freedo for the chi-sqare test was [ E ( E ) ri] which r i the nber of oveents is not passable dring the process. If the Markov chain has no stationary point it ay happen that the process occrs is seasonal. Different tie periods sed for each year of the transition atrix generated bt st se the sae tie interval. Ths, the nare of the still not fit to be ade se of divination bt nderlying factors inflencing the processes occrring. We can also find the Stationarity Markov (Markov specified) at each tie interval sed to see that process happen. We can increase the nber of tie intervals sch as Markov chain generated by a high-level bt still focs on the first stage of the ethod. Hypothesis for the Markov chain at each tie interval are shown below: T H ( ) : q t, t q and the Chi-Sqare wold be : n i, j n, T E E ni ( ) N i X 0 j t n i, j ni Ni Which the degree of freedos T EE ri. ii. Analysis probles This calclation is contined ntil we get the nber of insrance prchasing in 006. The s of the total prchase one, two and three insrance policies for each year are:

5 Predicting nber of prchasing life insrance X X X X Chi-sqare vales only take readings in 004, 005, 006 as the oveent of cstoer behavior. However, data acqisition policy in 003 is sed to obtain the probability of 004. Calclations are sarized in the table below: Table : Coparison of the vales obtained are statistically stationary and the chi-sqare distribtion Intervals X dof for 95% 99% Total All other vales indicate less than 95% and 99%. Therefore, the hypothesis of hoogeneos Markov chain has less tie. Interval of tie to explain the entire process of prchasing behavior can not be deterined becase of prchase behavior in each year is not fixed. However, alternative ethods have been sed to test the predictive capability of the odel. This ethod ses the coparison of the clative expected vale at the tie of prchase will coe with the acal vale of prchase. iii) Real Model and Silation Model

6 409 Mohd Rahiie Bin Md Noor and Zaidi at Isa Let n (t,) is the nber of registered cstoers and ake a prchase insrance at the tie interval t for year. Than n (t,) and n (t,3) is nber of cstoers who by two and three policy policy for the first tie in the year. Tie 0 is the beginning of the existing data. Model predictions for the nber of prchases dring the period of insrance ( c year) is based on the clative nber of cstoers c ( t, i) who akes a prchase at the tie t. Here is the eqation sed in the odel forecasts for the prchase of fre policy c c c t, c t, q n ( t,) t, c t, q c t,q n t, t,3 c t, q c t,q c t,3 n t, 3 3 Forecast at the tie t of prchase of insrance also st satisfy the following eqation C t 0c t, c t, c t,3. These forlas are derived fro the assption cstoers by an insrance policy. The vales obtained will be coparable with the acal odel. However, the nber of policies sed in the acal odel is two and nber three prchases only and are derived fro annal data. Reslts obtained between odel predictions and acal odels shown below: Table : Nber of cstoers who by two and three sppleental insrance policy every year. Intervals Year Prediction Model Acal Model Accracy Percentge

7 Predicting nber of prchasing life insrance 4093 Figre : Coparison of acal vale insrance prchase forecasting odel iv Conclsion We can expect siilar vales if sing Markov test the second stage. At this stage, the second principle of the Markov chain is seen to narrow the scope of a process saple. So, this siation will becoe ore relevant engkin if saller saples sed. We also need to take into accont the nber of cstoers who by this policy onthly nbers. Nber of cstoers will ake ore accrate prediction odels becase we can see the changes that occr on a onthly basis.

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