Example.
The number of claims per year is Poisson with mean 3. The size of claims is Weibull with
= 0.5 and = 1000.
You are given the following uniform random numbers from [0, 1] to simulate the times between claims
0.52
0.29
0.78
0.37
0.69
and the follo

Credibility
Motivation
In a large portfolio of insurance policies, there is a subgroup of policies such that the
average cost for the group is different from that of the total portfolio, and the members
of the subgroup have the same characteristics. Or t

Simulation
Idea of Simulation
Inversion Method
Let FX (x) be the distribution function of X, and let u be a random number from the
uniform distribution over (0, 1).
If F
1
1
(u) exists, then x = F
(u).
If F (c ) u < F (c), then x = c.
If F
1
(u) = [a

Parametric Models
Moments and Percentile Matching
Suppose n independent observations are from the same parametric distribution with
distribution function F (x|), here = (1 , . . . , p ) is the parameter vector.
0k = (xk1 + + xkn )/n
Let 0k () be the kth

Final Review Problems
1.
You are given:
The number of claims has a Poisson distribution.
Claim sizes have a Pareto distribution with parameters = 0.5 and = 6.
The number of claims and claim sizes are independent.
The observed pure premium should be wi

Problem Set 1 (Part 3)
1.
Name
A sample of 3000 policies contained 1000 with no claims, 1200 with one claim, 600 with
two claims, and 200 with three claims. Use mle and a normal approximation to construct
a 90% confidence interval for the mean of a Poisso

Problem Set 6
1.
You are to simulate four observations from a binomial distribution with two trials and
probability of success 0.3, using the following random numbers from the uniform distribution on [0, 1]: 0.91, 0.21, 0.72, 0.48. How many observations o

Some Homework 2 Solutions
Name
3.22) Let X1 , X2 , . , and X100 be the 100 claims. We need E(X1 ) and E(X12 ). From our
formula sheet, these values are 4800 and 115, 200, 000, respectively. This gives us
Var(X1 ) = 92, 160, 000
1 = 9600
We let S =
100
X
X

Some Homework 1 Solutions
2.3) We need to solve the equation f 0 (x) = 0, so we need to take a first derivative.
f 0 (x) =4(1 + x2 ) 3 + 4x ( 3)(1 + x2 )
4(1 + x2 )
24x2
=
(1 + x2 )4
(1 + x2 )4
=4 20x2 (1 + x2 )4
4
2x
1
It follows that x = p .
5
2.4) We

Some Homework 3 Solutions
Solutions to 5.11, 6.3, 8.2T, 8.4, 8.5, 3.1, 3.2, 3.3, and 3.4.
2
5.11) For the rst distribution, HQ) 2 (2
)(.5)2. For the second distributino P2(2) =
4
(2) (.5)2(.5)2. (Im using capital P because we have a lowercase p that is an

Problem Set 1
1.
Name
A sample of size 5 produced the values 4, 5, 21, 99, and 421. You fit a Pareto distribution
using the method of moments. Determine the 95th percentile of the fitted distribution.
Answer
2.
Losses have a Burr distribution with = 2. A

Problem Set 5
1.
Name
There are two classes of risk for an insurance coverage. Loss amounts for each class are
as follows:
Class A, 90 risks
Loss Size Probability
100
0.8
300
0.2
Class B, 10 risks
Loss Size Probability
100
0.7
400
0.3
Determine B
uhlmanns

Problem Set 4
1.
Name
The average claim size for a group of insureds is 1500 with a standard deviation of
7500. Assume that claim counts have the Poisson distribution. Determine the expected
number of claims so that the total loss will be within 6% of the

Problem Set 5, Part 2
1.
Name
Three individual policyholders were observed for four years. Policyholder X had claims
of 2, 3, 3, and 4. Policyholder Y had claims of 5, 5, 4, and 6. Policyholder Z had claims
of 5, 5, 3, and 3. Use nonparametric empirical B

Problem Set 1 (Part 2)
1.
Name
2
Four observations were made from a random variable having density f (x) = 2xex ,
x > 0. Exactly one of the four observations was less than 2.
= 1 ln 4 .)
(a) Determine the mle of . (In Part 1, we had
4
3
(b) Approximate

Math 5634: Loss Models II
Instructor
Office
E-mail
Phone
Office Hours
Chunsheng Ban
MA 210
[email protected]
614-292-5331
4:30-5:20 p.m. Tuesdays; 3:50-4:30 p.m. Thursdays
and by appointment
COURSE DESCRIPTION
The two-semester sequence Math 5633/5634 introdu

Test 2 Topics
Hypothesis Testing
2 test
Likelihood ratio test
SBC
Credibility
Limited fluctuation credibility
Full and partial credit, partial credit premium
Bayesian estimate
Credibility premium (linear) estimate
B
uhlmann credibility
Sample Test

Problem Set 2
1.
Name
Given that the parameter is uniform over [0, 1] and P(Y = y|) = (y + 1)2 (1 )2 .
Calculate the unconditioonal probability P(Y = y).
Answer
2.
The prior distribution of H is P(H = 1/4) = 4/5 and P(H = 1/2) = 1/5. The
observation from

Test 1 Topics
Moment and Percentile Matching
Maximum Likelihood
Individual, grouped, and modified data
Variance of MLE
Fishers information, interval estimate
Delta Method
Discrete Models
Bayesian Method
Sample Test 1
1.
Name
The number of claims for

Problem Set 4
1.
Name
The average claim size for a group of insureds is 1500 with a standard deviation of
7500. Assume that claim counts have the Poisson distribution. Determine the expected
number of claims so that the total loss will be within 6% of the

Problem Set 3
1.
Name
Five observations are made: 1, 2, 3, 5, 13. Determine the Kolmogorov-Smirnov test
statistic for the null hypothesis that the density is f (x) = 2x2 e2/x , x > 0.
Answer
2.
You are given the following:
You have segregated 25 loses in

Some Homework 4 Solutions
Name
8.6) With our earlier observation about this distribution (from the previous solution set),
we can use all our formulas for an exponential distribution with an adjustment of .3. As
a reminder, this distribution is a mixture