Brosius Flashcards
When a<0
Our estimate of y will be negative for small values of X, use link ratio method instead
When b<0
Our estimate of y decreases as x increases, use budgeted loss instead
Hugh White’s question
When reported portion of expected losses is higher than expected: -Reduce IBNR (BL) -Keep IBNR the same (BF) -Increase IBNR (link ratio)
Options for using the link ratio method for poisson binomial
-Unbiased estimate -Minimize the MSE -Use E(Y/X) for c -Salzmann’s iceberg technique
General coefficient trends
a: Decrease over time c(LDF): Decrease over time d(1/c): Increase over time (always
What is an advantage of the least squares method?
Flexible because it gives more or less weight to the observed value of X as appropriate (i.e., credibility weighting)
List the 3 special cases for Brosius least squares
Chainladder -> y=bx -> a=0 BF -> y =a+x -> b=1 y=a -> Budgeted loss -> b=0
If claim counts are poisson(u), and d=probability of reporting in first period, what is the Bayes Reserve?
R(x)=u(1-d)
Negative binomial distribution
Y~negative binomial(r,p): E[Y]=r(1-p)/p
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?
R(x) = s/(1-s)(x+r) s=(1-d)(1-p)
What are the formulas for a and b in the Brosius Least Squares?
b= (mean(xy) - mean(x)mean(y))/(mean(x^2)-mean(x)^2) a=mean(y)-bmean(x)
For ultimate losses Y, and reported losses X, what is the best linear estimate of y given x?
L(x)=[x-EX]Cov(X,Y)/Var(x)+EY
When is Brosius Least Squares appropriate? Inappropriate?
Appropriate: When the distribution is the same across multiple years Inappropriate: Year to year changes are due to systemic shifts, eg. inflation, legal environment
Estimate ultimate losses L(x) using a credibility formula
L(x)=Zx/d+(1-Z)EY Z=VHM/(VHM+EVPV) EVPV=Ey(Var(X/Y)) VHM=Vary[E(X/Y)]
Calculate the statistics used in Brosius
c=mean(y)/mean(x)-> LDF Z=b/c -> Credibility d= 1/c -> % Reported
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.
⇧ 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)
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.
⇧ 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
Give three advantages of using the best linear approximation to Q as a replacement for the pure Bayesian estimate.
⇧ Simpler to compute ⇧ Easier to understand and explain ⇧ Less dependent upon the underlying distribution
Briefly describe a situation in which it would be appropriate to use the following methods: a) Link ratio method
Use the link ratio method for older accident years where loss development is stable
Briefly describe a situation in which it would be appropriate to use the following methods: b) Budgeted loss method
Use the budgeted loss method when past data is not available
Briefly describe a situation in which it would be appropriate to use the following methods: c) Least squares method
Use the budgeted loss method when past data is not available
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
It requires knowledge of the loss and loss reporting processes
