stat 477 name: winter 2016 final exam

Stat 477 Name:Winter 2016Final Exam16 April 2016

This exam contains 16 pages (including this cover page)and 8 problems. Check to see if any pages are missing.

You may only use an SOA-approved calculator and apencil or pen on this exam.

You are required to show your work on each problem onthis exam.

Problem Points Score

1 11

2 24

3 13

4 11

5 10

6 15

7 16

8 0

Total: 100

Stat 477 Final Exam - of 16 16 April 2016

1. You are given:

• N is the number of claims, and has a Poisson distribution with mean 3

• X is the size of an individual claim and has the following distribution:

X Pr(X)

500 2/31100 1/62100 1/6

• N and X are independent random variables

• S is the aggregate claim amount,

S =N∑i=1

Xi

(a) (3 points) Derive the variance of S.

(b) (4 points) Find the probability that S = 3200.

(c) (4 points) Find the probability that S > 2000.


Mostly blank page to be used as extra space if needed.


2. Assume you observe a sample of claim frequencies from a book of business.

Claims Policyholders

0 3001 2002 303 5

4+ 0

(a) (6 points) Assume that the data follow a zero-modified Poisson(2) distribution. Derivethe maximum likelihood estimate of p, the probability of zero (the final answer, 0.5607, isin the tables).

Assume now that the claims above come from a Poisson(λ) distribution, and the prior distri-bution for λ is Gamma(2,3).

(b) (7 points) Given the data, what is the probability that a new policyholder from this bookhas at least two claims?

(c) (6 points) What is the posterior distribution of λ?

(d) (5 points) What is the greatest accuracy credibility estimate of the number of claims froma policyholder in this book of business?


3. The following 10 observations are a random sample of claims from a block of business.

1, 2, 2, 13, 13, 13, 17, 17, 35, 100

(a) (3 points) Using a triangular kernel with bandwidth 2, find S(34).

(b) (3 points) Using a uniform kernel with bandwidth 1, find f(15).

(c) (3 points) Using a uniform kernel with bandwidth 100, find f(15).

(d) (4 points) What bandwidth on a uniform kernel would maximize the value of f(15)?


4. You are given the following information from a survival study of people with a certain disease(all numbers in months):

Observation First Last Status at LastNumber Observation Observation Observation

1 0 3 Dead2 0 4 Dead3 0 9 Dead4 0 13 Alive5 0 18 Alive6 0 19 Dead7 0 20 Alive8 0 20 Alive9 0 20 Alive10 2 9 Dead11 5 9 Dead12 8 20 Alive13 13 18 Alive14 17 20 Alive15 17 20 Alive

(a) (4 points) Find S(15) using the Kaplan-Meier estimate.

(b) (4 points) Find S(15) using the Nelson-Aalen estimate.

(c) (3 points) Assuming that the survival time of people with this disease has an Exponentialdistribution with a mean of θ, what is the total likelihood contribution of observations 6and 13?


5. (10 points) For a given policy, there are two types of policy holders. The loss amount of apolicy holder from type one is exponentially distributed with a mean of 2. Type 2 losses areuniformly distributed between 0 and 40.

X1 ∼ Exp(2) X2 ∼ Unif(0, 40)

Also, assume that S = X1 + X2, the total loss from a portfolio of one policyholder of type 1and one of type 2 (the policyholders are independent). Show that VaR0.9 = 38. (You will notbe able to solve the last equation without a computer. Just plug in 38 to check)


6. A random sample of payments (X) from a portfolio of policies resulted in the following:

Interval # of Policies

[0, 50] 50(50, 250] 20(250,1000] 7(1000,∞) 3

Claim sizes are uniformly distributed within each interval. Calculate the following, if possible.

(a) (3 points) Find E(X)

(b) (4 points) Find E(X ∧ 175)

(c) (4 points) Find E(X − 175)+

(d) (4 points) Using the ogive, find S(175).


7. Assume that the number of claims from a policy holder follows a Poisson distribution withmean λ. Further assume that λ follows a Gamma(3,7) distribution. In the first ten years, apolicyholder has 1 claim the first year, 2 the second year, and 0 the third through tenth years.

(a) (8 points) Calculate the Buhlmann premium for the eleventh year.

(b) (8 points) Calculate the Bayesian premium for the eleventh year.


8. (5 points (bonus)) Identify the following five members of our class. If your picture is one ofthe first five, identify the sixth picture.

stat 477 name: winter 2016 final exam

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