Brosius Flashcards
what fluctuations is real-world loss data subject to?
Brosius - Least Squares
- random fluctuations
2. systematic distortions
how might results be stabilized?
Brosius - Least Squares
developed losses can be weighted with a prior estimate
when is the least squares method good to use?
Brosius - Least Squares
when random year to year fluctuations in loss experience are significant
in the link ratio method [L(x) = c * x], when is choice of c challenging?
(Brosius - Least Squares)
when the link ratios vary significantly from year to year
ex. thin data, major fluctuations
when might the value of x be ignored (as in the budgeted loss method) when considering L(x)?
(Brosius - Least Squares)
- fluctuation in loss experience is extreme
2. past data is not available
how is c chosen for link ratio method?
Brosius - Least Squares
observed from previous AYs, calculated as a weighted avg. of several years
how is k chosen for budgeted loss method?
Brosius - Least Squares
averaging y over several years, or by multiplying earned premium by an expected loss ratio (i.e. expected losses)
how does the least squares method estimate L(x)?
Brosius - Least Squares
fits a line to the points (x, y) using the method of least squares
what is the special case of the least squares method where a = 0?
(Brosius - Least Squares)
link ratio method
what is the special case of the least squares method where b = 0?
(Brosius - Least Squares)
budgeted loss method
what is the special case of the least squares method when b = 1?
(Brosius - Least Squares)
BF method - sets L(x) = a + x for some a
what is an important advantage of the least squares method?
Brosius - Least Squares
flexibility - gives more or less weight to observed value of x as appropriate (i.e. credibility-weighting)
what are two types of parameter estimation errors that can lead to unrealistic a and b values?
(Brosius - Least Squares)
- significant changes in the nature of the loss experience
2. sampling error
problem and solution when a<0 ?
Brosius - Least Squares
problem: estimate of y will be negative for small values of x
solution: use link ratio method
problem and solution when b<0 ?
Brosius - Least Squares
problem: estimate of y decreases as x increases
solution: use budgeted loss method
what is notable about the graph produced by the link ratio method?
(Brosius - Least Squares)
fits a line through the origin
what is notable about the graph produced by the budgeted loss method?
(Brosius - Least Squares)
fits a horizontal line
what is notable about the graph produced by the least squares method?
(Brosius - Least Squares)
yields a “best fit” line - probably not through the origin
where do the lines from the link ratio, budgeted loss, and least squares method intersect?
(Brosius - Least Squares)
at the point (x_bar, y_bar)
if the reported proportion of expected losses as of the statement date for the current AY is 8% higher than it should be, how would the budgeted loss method respond?
(Brosius - Least Squares)
reduce the bulk reserve by a corresponding amount
assumes that losses were just reported more quickly than expected, overall loss amount will be same
if the reported proportion of expected losses as of the statement date for the current AY is 8% higher than it should be, how would the B/F method respond?
(Brosius - Least Squares)
leave the bulk reserve at the same percentage level of expected losses
(assumes ultimate losses will be higher by less than 8% - reserves will stay same)
if the reported proportion of expected losses as of the statement date for the current AY is 8% higher than it should be, how would the link ratio method respond?
(Brosius - Least Squares)
increase the bulk reserve in proportion to the increase of actual reported over expected reported
(assumes that ultimate losses will be 8% higher)
which method best describes Q(x) and R(x) for the case where Y ~ U(0,1) and X~Bin
(Brosius - Least Squares)
Least Squares method
which method best describes Q(x) and R(x) for the general Poisson-binomial case?
(Brosius - Least Squares)
Bornhuetter/Ferguson
Why does the Bornhuetter/Ferguson best describe Q(x) and R(x) for the general Poisson-binomial case?
(Brosius - Least Squares)
- link ratio method doesn’t work because there’s no c such that x + mu * (1 - d) = c * x
- expected number of outstanding claims, R(x) doesn’t depend on the number of claims already reported
what are four ways to use the link ratio method to describe the Poisson-Binomial case of Q(x) and R(x)?
(Brosius - Least Squares)
1 - Unbiased estimate - set E[(c - 1)X] = R(x)
2- minimize the MSE (E[((c - 1) X - R(x))^2] (biased low)
3- Use E{Y/X] for c (problematic when data is thin - undefined for X = 0) (biased high)
4- Salzmann’s “iceberg” technique - set E{X/Y | Y != 0] = d, c = 1/d
why don’t the budgeted loss, BF, and link ratio methods describe Q(x) and R(x) for the negative binomial-binomial case?
(Brosius - Least Squares)
budgeted loss and BF - except in the trivial case where d = 1, it’s an increasing linear function of x - so an increase in reported claims leads to an increase in our estimate of outstanding claims
link ratio - relationship is not proportional
why does an increasing function for R(x) make sense for the negative binomial-binomial case?
(Brosius - Least Squares)
NB has more variance than the Poisson with the same mean -> we have less confidence in our prior estimate of expected losses and are more willing to increase our estimated ultimate claim count
what method best describes the fixed prior case for Q(x) and R(x)?
(Brosius - Least Squares)
budgeted loss method
what method best describes the fixed reporting case for Q(x) and R(x)?
(Brosius - Least Squares)
link ratio method
why is it difficult to compute a pure Bayesian estimate Q?
Brosius - Least Squares
requires knowledge of the loss and loss reporting processes, which make it difficult to make assumptions
what are the advantages of using the best linear approximation as a replacement for the Bayesian estimate?
(Brosius - Least Squares)
1- simpler to compute
2- easier to understand and explain
3- less dependent upon the underlying distribution
via the best linear approximation L(x) to Q, when would a large reported amount lead to a DECREASE in the reserve?
(Brosius - Least Squares)
if Cov(X, Y) < Var(X) (budgeted loss method)
via the best linear approximation L(x) to Q, when would a large reported amount NOT AFFECT the reserve?
(Brosius - Least Squares)
Cov(X, Y) = Var(X)
BF method
via the best linear approximation L(x) to Q, when would a large reported amount lead to an INCREASE in the reserve?
(Brosius - Least Squares)
Cov(X, Y) > Var(X)
link ratio method
when is least squares fit inappropriate?
Brosius - Least Squares
if year to year changes in loss experience are due largely to systematic shifts or distortions in the book of business
when is least squares fit appropriate?
Brosius - Least Squares
if year to year changes are largely due to random chance
two examples of how data can be adjusted for systematic distortions before applying the least squares method?
(Brosius - Least Squares)
1- if studying incurred loss data, correct for inflation by putting the years on a constant-dollar basis before fitting a line
2- if business expands, divide each year’s losses by an exposure base to eliminate the distortion
what does Expected Value of the Process Variance (EVPV) represent?
variability resulting from the loss reporting process
what does the Variance of the Hypothetical Mean (VHM) represent?
variability resulting from the loss occurrence process
what is L(x) = Z * x/d + (1 - Z) * E[Y] doing?
Brosius - Least Squares
credibility-weighting of the link ratio estimate (x/d) and budgeted loss estimate (E[Y])
what is the result when EVPV = 0 in the L(x) credibility mixture of link ratio and budgeted loss?
(Brosius - Least Squares)
full weight given to the link ratio estimate (fixed reporting)
what is the result when VHM = 0 in the L(x) credibility mixture of link ratio and budgeted loss?
(Brosius - Least Squares)
full weight given to the budgeted loss estimate (fixed prior)
what are two benefits of the least squares method?
Brosius - Least Squares
- easy to implement
2. uses easily accessible data
what does the least squares method work well for?
Brosius - Least Squares
developing losses for small states or lines that are subject to serious fluctuations
what are two possible issues/errors with the least squares method?
(Brosius - Least Squares)
- errors if corrections aren’t made to account for significant exposure changes or other shifts in loss history
- subject to sampling error since parameters are estimated from observed data
