INSTITUTE OF ACTUARIES OF INDIA

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1 INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 9 h November 2010 Subjec CT6 Saisical Mehods Time allowed: Three Hours ( Hrs.) Toal Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1. Please read he insrucions on he fron page of answer bookle and insrucions o examinees sen along wih hall icke carefully and follow wihou excepion 2. Mark allocaions are shown in brackes. 3. Aemp all quesions, beginning your answer o each quesion on a separae shee. However, answers o objecive ype quesions could be wrien on he same shee. 4. In addiion o his paper you will be provided wih graph paper, if required. AT THE END OF THE EXAMINATION Please reurn your answer book and his quesion paper o he supervisor separaely.

2 Q. 1) A veeran acuary believes ha he claims from a paricular ype of policy follow he Burr disribuion wih parameers 2, 1000 and As per his recommendaion, he insurance company has se a deducible such ha 25% of he losses resul in no claim o he insurer. (i) (ii) Calculae he size of he deducible. An acuarial rainee suspecs ha he deducible se by he veeran acuary is based on more of surmise han daa. She has access o daa on 1250 claims (ne of deducible). Coninuing wih he assumpion of he Burr disribuion for he original claims, she wishes o esimae is parameers from he available daa, by using he mehod of maximum likelihood. Give an expression for he probabiliy densiy funcion of he observed daa (ne of deducible), and he likelihood funcion ha has o be maximized. (iii) Give an expression for he maximum likelihood esimae (MLE) of he rue fracion of he losses ha resul in no claim o he insurer, in erms of he MLE of he parameers. (3) (4) (2) Q. 2) The annual number of claims on a paricular risk has he Binomial disribuion wih maximum claim number 10 and average claim number. The prior densiy of he n1 n2 n n 1! parameer is 1, where n 1 and n 2 are known posiive 10 n1! n2! inegers. The number of claims in he years 2007, 2008 and 2009 were X 1, X 2 and X 3, respecively. (i) Deermine he prior mean of. (2) (ii) Deermine he maximum likelihood esimaor of. (2) (iii) Deermine he Bayes esimae of he number of claims in he year 2010, under he squared error loss funcion. (4) (iv) Show ha he esimaor of par (iii) has he form of a credibiliy esimae, and idenify he credibiliy facor. (2) (v) Deermine he credibiliy esimaor of under EBCT Model 1 and compare wih he resul of par (iii). (6) Q. 3) The aggregae claims process for a risk is a compound Poisson process wih rae 50 per annum. Individual claim amouns are Rs wih probabiliy 0.25, Rs wih probabiliy 0.5, or Rs wih probabiliy The premium loading is 10%. Le S denoe he aggregae annual claim amoun. (i) Calculae he mean and variance of S. (2) (ii) Using a normal approximaion o he disribuion of S, calculae he iniial surplus required in order ha he probabiliy of ruin a he end of he firs year is (3) (iii) A reinsurer offers o sell o he insurer proporional reinsurance for 25% of he claims, for premium loading 15%. If his offer is acceped, calculae he modified iniial surplus required in order ha he probabiliy of ruin a he end of he firs year is (4) Page 2 of 5 [16]

3 Q. 4) The cumulaive incurred claims (in housands of rupees) on a porfolio of insurance policies are as given in he following able. Acciden Year Developmen Year ,463 2,749 3,529 3, ,013 3,278 3, ,321 3, ,953 The earned premium for he year 2009 is Rs. 6,472,000, while he paid claims are Rs. 1,731,000. (i) Assuming ha he Ulimae Loss Raio is 88%, calculae he reserve needed for 2009 using he Bornhueer-Ferguson (basic) mehod. (8) (ii) Sae he assumpions underlying he use of he above mehod. (3) Q. 5) Consider he auoregressive process given by Y Y Z, 2 2 Z being whie noise wih mean zero and variance. (i) Wha is he range of values of he real valued parameer so ha he process is saionary? (2) (ii) Obain a represenaion of Y as a Z, by specifying a j j 0, a 1, explicily. (3) j 0 (iii) Using par (b) or oherwise, find an expression for he variance of Y in erms of 2 and. (2) (iv) Compare he resul of par (iii) wih he variance of an AR(1) process and explain any similariy or dissimilariy. (2) Q. 6) The sample ACF and PACF values a lags 1 o 10 of a ime series of lengh 500, are as given below. Lag SACF SPACF (i) Deermine hrough a saisical es wheher he ime series can be regarded as whie noise. (5) (ii) Indicae, wih reasons, if an AR(p) or an MA(q) model may be appropriae for his ime series, and if so, wha could be he model order. (4) Page 3 of 5 [11]

4 Q. 7) Lis six perils ha are ypically insured agains under a household building policy. [3] Q. 8) A claim analys of a healh insurance company examines daa on a porfolio of healh insurance policies. He plans o use a generalized linear model for he claim amouns, involving he following raing facors. SA : Sum assured (x), AG : Age group, OC : Occupaion, a coninuous variable. a facor wih 10 levels. a facor wih 6 levels. A preliminary analysis produces he following summary for he models considered by he analys. Model Linear predicor No of parameers Scaled deviance SA x SA + AG?? SA + AG + SA * AG?? SA * AG + OC?? SA * AG * OC?? 58.9 (i) Complee he able by filling in he cells wih quesion marks. (4) (ii) On he basis of he scaled deviance, which model should he analys choose? (3) (iii) Wha furher consideraions should be given before he analys makes a recommendaion abou he choice of he model? (2) Q. 9) An acuary uses he following algorihm o generae pseudorandom numbers X from he Poisson disribuion wih mean. Sep 1: Inpu lambda. Sep 2: Se X=0; Z=0. Sep 3: Se Y = Random sample from he uniform U(0,1) disribuion. Sep 4: Incremen Z by he amoun ln(y)/lambda. (ln is he log funcion). Sep 5: If Z<1, hen incremen X by 1; GO TO Sep 3. Sep 6: Oupu X. Sep 7: GO TO Sep 2 for generaing he nex value of X. (i) By analysing he above algorihm, show ha i generaes he value X 0 wih he correc probabiliy. (3) (ii) If five successively random samples from he uniform disribuion, generaed in Sep 3, happen o be 0.564, 0.505, 0.756, and 0.046, and 2, follow he above algorihm o generae as many samples of X as his informaion permis. (4) (iii) Using he uniformly disribued random samples and he value of given in par (ii), generae samples (as many as possible) from he Poisson disribuion, using he Page 4 of 5

5 inverse disribuion ransform mehod. (5) [12] Q. 10) (i) Sae he individual risk model, wih a clear descripion of he assumpions. (5) (ii) How is his model differen from he collecive risk model? (3) Q. 11) The owner of a personal compuer has o decide wheher o sign an Annual Mainenance Conrac (AMC) or o pay for repair separaely on each occasion of compuer faul. The AMC coss Rs. 1000, and provides for an unlimied coun of repair services. In he absence of he AMC, he servicing agency charges Rs. 300 for each repair. The owner assumes he probabiliy disribuion of he annual number of fauls as follows. Number of fauls More han 5 Probabiliy (i) Form he loss marix for he owner of he compuer in respec of he above decision. (2) (ii) Wha is he minimax decision? (1) (iii) Wha is he Bayes decision? (2) ************************ [8] [5] Page 5 of 5