Brosius

Question 1

When a<0

Answer 1

Our estimate of y will be negative for small values of X, use link ratio method instead

Question 2

When b<0

Answer 2

Our estimate of y decreases as x increases, use budgeted loss instead

Question 3

Hugh White’s question

Answer 3

When reported portion of expected losses is higher than expected: -Reduce IBNR (BL) -Keep IBNR the same (BF) -Increase IBNR (link ratio)

Question 4

Options for using the link ratio method for poisson binomial

Answer 4

-Unbiased estimate -Minimize the MSE -Use E(Y/X) for c -Salzmann’s iceberg technique

Question 5

General coefficient trends

Answer 5

a: Decrease over time c(LDF): Decrease over time d(1/c): Increase over time (always

Question 6

What is an advantage of the least squares method?

Answer 6

Flexible because it gives more or less weight to the observed value of X as appropriate (i.e., credibility weighting)

Question 7

List the 3 special cases for Brosius least squares

Answer 7

Chainladder -> y=bx -> a=0 BF -> y =a+x -> b=1 y=a -> Budgeted loss -> b=0

Question 8

If claim counts are poisson(u), and d=probability of reporting in first period, what is the Bayes Reserve?

Answer 8

R(x)=u(1-d)

Question 9

Negative binomial distribution

Answer 9

Y~negative binomial(r,p): E[Y]=r(1-p)/p

Question 10

Claim counts are negbin(r,p), d is the probability of reporting in the first period, x is the actual reoprted in the first period. What is the Bayes Reserve?

Answer 10

R(x) = s/(1-s)(x+r) s=(1-d)(1-p)

Question 11

What are the formulas for a and b in the Brosius Least Squares?

Answer 11

b= (mean(xy) - mean(x)mean(y))/(mean(x^2)-mean(x)^2) a=mean(y)-bmean(x)

Question 12

For ultimate losses Y, and reported losses X, what is the best linear estimate of y given x?

Answer 12

L(x)=[x-EX]Cov(X,Y)/Var(x)+EY

Question 13

When is Brosius Least Squares appropriate? Inappropriate?

Answer 13

Appropriate: When the distribution is the same across multiple years Inappropriate: Year to year changes are due to systemic shifts, eg. inflation, legal environment

Question 14

Estimate ultimate losses L(x) using a credibility formula

Answer 14

L(x)=Zx/d+(1-Z)EY Z=VHM/(VHM+EVPV) EVPV=Ey(Var(X/Y)) VHM=Vary[E(X/Y)]

Question 15

Calculate the statistics used in Brosius

Answer 15

c=mean(y)/mean(x)-> LDF Z=b/c -> Credibility d= 1/c -> % Reported

Question 16

You are trying to establish the reserve for commercial auto bodily injury, and the reported proportion 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.

Answer 16

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

Question 17

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.

Answer 17

⇧ 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

Question 18

Give three advantages of using the best linear approximation to Q as a replacement for the pure Bayesian estimate.

Answer 18

⇧ Simpler to compute ⇧ Easier to understand and explain ⇧ Less dependent upon the underlying distribution

Question 19

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

Answer 19

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

Question 20

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

Answer 20

Use the budgeted loss method when past data is not available

Question 21

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

Answer 21

Use the budgeted loss method when past data is not available

Question 22

Answer 22

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

Question 23

Answer 23

It requires knowledge of the loss and loss reporting processes