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1

Briefly describe three differences between Hurlimann's method and the Benktander method

Hurlimann
1) Hu ̈rlimann’s method is based on a full development triangle, whereas the Benktander method is based on a single accident year. 2) Hu ̈rlimann’s method requires a measure of exposure for each accident year (i.e. premiums) 3) Hu ̈rlimann’s method relies on loss ratios (rather than link ratios) to determine reserves

2

Briefly describe one similarity between Hu ̈rlimann’s method and the Benktander method.

Hurlimann
Similar to the Benktander method, Hu ̈rlimann’s method represents a credibility weight- ing between two extreme positions: relies on cumulative paid claims (i.e. individual loss reserves) vs. ignores cumulative paid claims (i.e. collective loss reserves)

3

Provide one advantage of the collective loss ratio reserve over the standard Bornhuetter/Ferguson reserve.

Hurlimann
With the collective loss ratio reserve, different actuaries always come to the same results provided they use the same premiums

4

Explain why t* = sqrt(p) is an appealing choice when calculating the optimal credibility weights.

Hurlimann
This assumption yields the smallest credibility weights for the individual loss reserves, which places more emphasis on the collective loss reserves

5

Briefly describe a situation in which it would be appropriate to use the following methods: Link ratio method

Brosius
Use the link ratio method for older accident years where loss development is stable

6

Briefly describe a situation in which it would be appropriate to use the following methods: Budgeted loss method

Brosius
Use the budgeted loss method when past data is not available

7

Briefly describe a situation in which it would be appropriate to use the following methods: Least squares method

Brosius
Use the least squares method when random year to year fluctuations in loss experience are significant

8

You are trying to establish the reserve for commercial auto bodily injury, and the reported propor- tion of expected losses as of the statement date for the current accident year period is 8% more than it should be. Give three possible solutions for managing the bulk reserves. For each solution, identify a corresponding loss development method.

Brosius
1) Reduce the bulk reserve by a corresponding amount (budgeted loss method) 2) Leave the bulk reserve at the same percentage level of expected losses (Bornhuetter/Ferguson method) 3) Increase the bulk reserve in proportion to the increase of actual reported over expected reported (link ratio method)

9

An actuary is reviewing the incurred loss experience for a business with a growing book. In addition to the standard reserving methods, she would like to apply the least squares method. Briefly describe two adjustments that should be made to the data before applying the least squares method.

Brosius
Since she is reviewing incurred loss data, she can correct for inflation by putting the years on constant-dollar basis. Since the book is growing, she can divide each year’s losses by an exposure base to eliminate the distortion

10

Given the following: Y = number of claims incurred each year, X = number of claims reported by year-end, Q(x) = E[Y |X = x]. Briefly explain why it is difficult to compute Q.

Brosius
It requires knowledge of the loss and loss reporting processes

11

Given the following: Y = number of claims incurred each year, X = number of claims reported by year-end, Q(x) = E[Y |X = x]. Give three advantages of using the best linear approximation to Q as a replacement for the pure Bayesian estimate

Brosius
1) Simpler to compute 2) Easier to understand and explain 3) Less dependent upon the underlying distribution

12

Given the following: Y = number of claims incurred each year ~ Poisson random variable with mean μ. X = number of claims reported by year-end. d = probability of a claim being reported by year end. μ(1-d) is a constant m. Briefly explain why the Bornhuetter/Ferguson method is the most appropriate reserving method to use in this situation.

Brosius
In the situation described above, X is a binomial random variable with parameters (y, d). Since Y is Poisson distributed, we have a Poisson-binomial mixed process. Thus, R(x) = μ(1 d). Since R(x) does not depend on the number of claims already reported, the Bornhuetter/Ferguson method is the optimal method to use

13

Given the following: Y = number of claims incurred each year ~ Poisson random variable with mean μ. X = number of claims reported by year-end. d = probability of a claim being reported by year end. μ(1-d) is a constant m. If an actuary decided to use the link ratio method, provide four options for determining the link ratio c.

Brosius
Option 1: Unbiased estimate • SinceR(x)=E(Y X|X=x)=m, we must set E[(c-1)X]=m in order to obtain an unbiased estimate . Option 2: Minimize the mean squared error • Minimize E[((c - 1)X - m)2]. Option3: Use E[Y/X|X0] for c. Option 4: Salzmann's iceberg technique• UseE[X/Y|Y 0] for d. Thus, c=1/d