STAT 479 Test 2 Spring 2014 April 1, 2014

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1 TAT 479 Test pring 014 April 1, (5 points) You are given the following grouped data: Calculate F (4000) 5 using the ogive. Amount of claims Number of Claims 0 to to to 10, ,000 and above F 5 18 (500) F 5 3 (10, 000) F 5 (4000) is determined using linear interpolation under the ogive F 5 10, (4000) , ,

2 . For a warranty insurance policy, you are given the following frequency distribution: Number of Claims Probability You are also given the following severity distribution: Amount of Claim Probability Weibo Warranty Company buys stop loss insurance from pears top Loss Company which will pay for aggregate claims in excess of pears charges 15% of the Net top Loss Premium for this coverage. a. (10 points) Determine f (x). f (0) Pr( N 0) 0.05 f (1000) Pr( N 1) Pr( X 1000) (0.)(0.4) 0.08 f (000) Pr( N ) Pr( Both X 1000) Pr( N 1) Pr( X 000) (0.3)(0.4) (0.)(0.5) f (3000) Pr( N ) Pr( One X 1000 and Other X 000) Pr( N 1) Pr( X 3000) (0.3)()(0.4)(0.5) (0.)(0.35) 0.13 f (4000) Pr( N ) Pr( One X 1000 and Other X 3000) Pr( N ) Pr(Both X 000) (0.3)()(0.4)(0.35) (0.3)(0.5) f (5000) Pr( N ) Pr( One X 000 and Other X 000) (0.3)()(0.5)(0.35) f N Both X (6000) Pr( ) Pr( 3000) (0.3)(0.35)

3 b. (6 points) Calculate the amount that Weibo will pay to pears. E[( 4000) ] E[ ] E[ 4000] E[ ] E[ N] E[ X ] (0.5)(0) (0.)(1) (0.3)() (0.4)(1000) (0.5)(000) (0.35)(3000) (0.8)(1950) 1560 E [ 4000] (0.5)(0) (0.08)(1000) (0.098)(000) (0.13)(3000) ( )(4000) 1434 E[( 4000) ] E[ ] E[ 4000] Premium (1.5)(16) Alternatively, we can calculate it directly as: E[( 4000) ] ( )(0.055) ( )( ) 16 Premium (1.5)(16)

4 3. Gloria from Gong Consulting has been hired by the Purdue athletic department to determine the transfer rate of athletes in the Purdue basketball program. he gathers the following data for a five year period with all times in years: Player tart Date in tudy Termination Date Termination Reason Graduation 0 1 Flunked Out Graduation Transfer Transfer Graduation Transfer Transfer 9 5 End of tudy 10 5 End of tudy Illness 1 4 Transfer End of tudy End of tudy End of tudy Gloria decides to use the Kaplan Meier product limit estimator to estimate (3.7) 15. a. (7 points) Determine (3.7) 15 as calculated by Gloria. y s 3 4 r (3.7) b. (5 points) Calculate the variance of this estimate using the Greenwood Approximation. 1 Var[ 15(3.7)] (0.75) (11)(11 ) (1)(1 1)

5 4. Wenchu has selected the following sample of claims: Wenchu creates a continuous distribution from this sample using a kernel density estimator with a uniform kernel with a bandwidth of. a. (8 points) Calculate f (4.5) and F (4.5) produced by the kernel density model. y 1 4 1/10 x 6 5 1/10 3x /10 4x /10 5x /10 6 x /10 7 x /10 1 x 16 p x k (4.5) 1/ b 1/ 4 =0.5 for y 4.5 y and zero elsewhere y f (4.5) p k (4.5) p k (4.5) p k (4.5) 0(for y >6.5)= (0.1)(0.5)+(0.1)(0.5)+(0.3)(0.5)=0.15 K y 4.5 y (4.5) for y 4.5 y ; 1 for y<.5 and 0 for y>6.5 4 F p4 K4 p5 K5 p6 K6 y (4.5) (4.5) (4.5) (4.5) 0(for >6.5)= (0.1) +(0.1) +(0.3) =

6 b. (3 points) Calculate the mean and variance of the kernel density model. Mean of kernel density model = Mean of Empiral Distribution = 4 5 (3)(6) ()(7) b Variance of kernel density model = Variance of Empiral Distribution (3)(6) ()(7) (7.)

7 5. (10 points) Claims for comprehensive coverage offered by Chellberg Car Assurance Company are distributed as a Pareto distribution with 5 and 000. Chellberg s chief actuary, Devin, wants to create a discrete distribution for claims using a span of 500. Devin asks Emily to discretize the claims using the method of rounding. Let probability that Emily assigns to the range of (750, 150). ROUNDING f be the Devin asks Brandon to discretize the claims using the method of local moment matching where the discretized distribution will have the same mean as the Pareto distribution. Let MomentMatching f be the probability that Brandon assigns to the range of (750, 150). ROUNDING MomentMatching Calculate 1000( f f ). 5 5 ROUNDING f F(150) F(750) f MomentMatching E[ X 1000] E[X ( )] E[X ( )] Answer (1000)( )

8 6. (10 points) The Dai Dog Insurance Company provides life insurance on new born puppies. Dai wants to understand the expected claims that she will be paying during the next year. Dora who is the President of the Dai believes that large dogs and small dogs have different mortality and different amount of claims. he has sorted the Dai s policies into two portfolios. The following is the information on the portfolios: Portfolio Number Probability of Death of Policies during Next Year Distribution of Death Benefit mall Dogs 10, Uniform from 1000 to 500 Large Dogs 8, policies have a benefit of policies have a benefit of 3000 Dora decides that she wants to hold a reserve equal to E[ ] 1.5 Var() where is the random variable representing the aggregate claims to be paid during the next year. Calculate the reserve that Dora will hold. This question could be worked two ways. One way (the proper way) is to split the Large Dog portfolio into two portfolios since we know the actual split by count. If you treat the face as a distribution for large dogs as 3/8 are for 3000 and 5/8 are 3000, you get a slightly different answer of 3,653, Credit was given for either answer, but the one below is correct E [ ] (10,000)(0.08) (3000)(0.11)(1500) (5000)(0.11)(3000) 3,545, Var() (10, 000) (0.08) (0.08) (1 0.08) 1 (3000)(0.11)(1 0.11)(1500 ) (5000)(0.11)(1 0.11)(3000 ) 7, 470,35.00 Reserve 3,545, , 470, , 653,

9 7. The Dai Dog Insurance Company provides life insurance on new born puppies. The insurance is paid with a single premium so the only terminations that can occur are from death. Dai hires Kevin to study mortality of 100 dogs insured by Dai. The 100 dogs die as follows: Year of Death Number of Dogs a. (5 points) Calculate H (3) using the Nelson-Åalen estimator. y s r H (3) b. (5 points) Calculate the 80% linear confidence interval for H (3) Var( H (3)) Confidence Interval = (0.4339, ) c. (1 point) Estimate (3) using the Nelson-Åalen estimator. e e H (3) (3)

10 7. (CONTINUED) The Dai Dog Insurance Company provides life insurance on new born puppies. The insurance is paid with a single premium so the only terminations that can occur are from death. Dai hires Kevin to study mortality of 100 dogs insured by Dai. The 100 dogs die as follows: Year of Death Number of Dogs d. ( points) Calculate the unbiased estimator of (3). Number Alive after Time (3) e. ( points) Calculate the variance of the unbiased estimator of (3). Var [ (3)][1 (3)] (0.55)(1 0.55) ( 100(3)) f. ( points) Calculate the unbiased estimator of 3 q n n q n g. ( points) Calculate the variance of the unbiased estimator of 3 q Var [ q ][1 q ] (0.583)( ) 3 3 ( 3q) n 600

11 8. hihao draws the following sample from a uniform distribution on the range of (0, ): a. (1 point) Calculate the unbiased estimate of the mean. X b. ( points) Calculate the unbiased estimate of the variance of this distribution. i nx x (3)(583.33) x 13, n 1 31 c. (4 points) Calculate the mean square error of the estimate in Part a. if ME Var( X ) bias( X ) Var( X ) ( ) 1 Var X n 3 36 bias( X ) 0 since X is an unbiased estimator ME= 0 7,

12 9. (10 points) For a disability policy sold to lumberacks, the number of claims in a year is distributed as geometric distribution with a mean on 0.5. The amount of each claim under these policies is distributed as a gamma distribution with 4 and Li Insurance Company has 8000 policies in force on January 1, 014. Assuming the normal distribution, calculate the 90% confidence interval for the aggregate amount of claims in 014. E[ ] (8000) E[ N] E[X] (8000)[0.05][(4)(5000)] 80,000,000 Var( ) 8000 Var( N) E[ X ] E[ N] Var( X ) 8000 (0.5)(1 0.5) (4)(5000) (0.5)(4)(5000) 80, 000, 000, 000 Confidence Interval = 80, 000, , 000, 000,000 (77,47,388.51, 8,75,611.49)

1. The number of dental claims for each insured in a calendar year is distributed as a Geometric distribution with variance of

1. The number of dental claims for each insured in a calendar year is distributed as a Geometric distribution with variance of 7.315. The amount of each claim is distributed as a Pareto distribution with