Brosius

Question 1

what fluctuations is real-world loss data subject to?

Brosius - Least Squares

Answer 1

1. random fluctuations

2. systematic distortions

Question 2

how might results be stabilized?

Brosius - Least Squares

Answer 2

developed losses can be weighted with a prior estimate

Question 3

when is the least squares method good to use?

Brosius - Least Squares

Answer 3

when random year to year fluctuations in loss experience are significant

Question 4

in the link ratio method [L(x) = c * x], when is choice of c challenging?

(Brosius - Least Squares)

Answer 4

when the link ratios vary significantly from year to year

ex. thin data, major fluctuations

Question 5

when might the value of x be ignored (as in the budgeted loss method) when considering L(x)?

(Brosius - Least Squares)

Answer 5

1. fluctuation in loss experience is extreme

2. past data is not available

Question 6

how is c chosen for link ratio method?

Brosius - Least Squares

Answer 6

observed from previous AYs, calculated as a weighted avg. of several years

Question 7

how is k chosen for budgeted loss method?

Brosius - Least Squares

Answer 7

averaging y over several years, or by multiplying earned premium by an expected loss ratio (i.e. expected losses)

Question 8

how does the least squares method estimate L(x)?

Brosius - Least Squares

Answer 8

fits a line to the points (x, y) using the method of least squares

Question 9

what is the special case of the least squares method where a = 0?

(Brosius - Least Squares)

Answer 9

link ratio method

Question 10

what is the special case of the least squares method where b = 0?

(Brosius - Least Squares)

Answer 10

budgeted loss method

Question 11

what is the special case of the least squares method when b = 1?

(Brosius - Least Squares)

Answer 11

BF method - sets L(x) = a + x for some a

Question 12

what is an important advantage of the least squares method?

Brosius - Least Squares

Answer 12

flexibility - gives more or less weight to observed value of x as appropriate (i.e. credibility-weighting)

Question 13

what are two types of parameter estimation errors that can lead to unrealistic a and b values?

(Brosius - Least Squares)

Answer 13

1. significant changes in the nature of the loss experience

2. sampling error

Question 14

problem and solution when a<0 ?

Brosius - Least Squares

Answer 14

problem: estimate of y will be negative for small values of x
solution: use link ratio method

Question 15

problem and solution when b<0 ?

Brosius - Least Squares

Answer 15

problem: estimate of y decreases as x increases
solution: use budgeted loss method

Question 16

what is notable about the graph produced by the link ratio method?

(Brosius - Least Squares)

Answer 16

fits a line through the origin

Question 17

what is notable about the graph produced by the budgeted loss method?

(Brosius - Least Squares)

Answer 17

fits a horizontal line

Question 18

what is notable about the graph produced by the least squares method?

(Brosius - Least Squares)

Answer 18

yields a “best fit” line - probably not through the origin

Question 19

where do the lines from the link ratio, budgeted loss, and least squares method intersect?

(Brosius - Least Squares)

Answer 19

at the point (x_bar, y_bar)

Question 20

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)

Answer 20

reduce the bulk reserve by a corresponding amount

assumes that losses were just reported more quickly than expected, overall loss amount will be same

Question 21

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)

Answer 21

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)

Question 22

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)

Answer 22

increase the bulk reserve in proportion to the increase of actual reported over expected reported
(assumes that ultimate losses will be 8% higher)

Question 23

which method best describes Q(x) and R(x) for the case where Y ~ U(0,1) and X~Bin
(Brosius - Least Squares)

Answer 23

Least Squares method

Question 24

which method best describes Q(x) and R(x) for the general Poisson-binomial case?
(Brosius - Least Squares)

Answer 24

Bornhuetter/Ferguson

Question 25

Why does the Bornhuetter/Ferguson best describe Q(x) and R(x) for the general Poisson-binomial case?
(Brosius - Least Squares)

Answer 25

• 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

Question 26

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)

Answer 26

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

Question 27

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)

Answer 27

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

Question 28

why does an increasing function for R(x) make sense for the negative binomial-binomial case?
(Brosius - Least Squares)

Answer 28

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

Question 29

what method best describes the fixed prior case for Q(x) and R(x)?
(Brosius - Least Squares)

Answer 29

budgeted loss method

Question 30

what method best describes the fixed reporting case for Q(x) and R(x)?
(Brosius - Least Squares)

Answer 30

link ratio method

Question 31

why is it difficult to compute a pure Bayesian estimate Q?

Brosius - Least Squares

Answer 31

requires knowledge of the loss and loss reporting processes, which make it difficult to make assumptions

Question 32

what are the advantages of using the best linear approximation as a replacement for the Bayesian estimate?
(Brosius - Least Squares)

Answer 32

1- simpler to compute
2- easier to understand and explain
3- less dependent upon the underlying distribution

Question 33

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)

Answer 33

if Cov(X, Y) < Var(X)
(budgeted loss method)

Question 34

via the best linear approximation L(x) to Q, when would a large reported amount NOT AFFECT the reserve?
(Brosius - Least Squares)

Answer 34

Cov(X, Y) = Var(X)

BF method

Question 35

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)

Answer 35

Cov(X, Y) > Var(X)

link ratio method

Question 36

when is least squares fit inappropriate?

Brosius - Least Squares

Answer 36

if year to year changes in loss experience are due largely to systematic shifts or distortions in the book of business

Question 37

when is least squares fit appropriate?

Brosius - Least Squares

Answer 37

if year to year changes are largely due to random chance

Question 38

two examples of how data can be adjusted for systematic distortions before applying the least squares method?

(Brosius - Least Squares)

Answer 38

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

Question 39

what does Expected Value of the Process Variance (EVPV) represent?

Answer 39

variability resulting from the loss reporting process

Question 40

what does the Variance of the Hypothetical Mean (VHM) represent?

Answer 40

variability resulting from the loss occurrence process

Question 41

what is L(x) = Z * x/d + (1 - Z) * E[Y] doing?

Brosius - Least Squares

Answer 41

credibility-weighting of the link ratio estimate (x/d) and budgeted loss estimate (E[Y])

Question 42

what is the result when EVPV = 0 in the L(x) credibility mixture of link ratio and budgeted loss?
(Brosius - Least Squares)

Answer 42

full weight given to the link ratio estimate (fixed reporting)

Question 43

what is the result when VHM = 0 in the L(x) credibility mixture of link ratio and budgeted loss?
(Brosius - Least Squares)

Answer 43

full weight given to the budgeted loss estimate (fixed prior)

Question 44

what are two benefits of the least squares method?

Brosius - Least Squares

Answer 44

1. easy to implement

2. uses easily accessible data

Question 45

what does the least squares method work well for?

Brosius - Least Squares

Answer 45

developing losses for small states or lines that are subject to serious fluctuations

Question 46

what are two possible issues/errors with the least squares method?

(Brosius - Least Squares)

Answer 46

1. errors if corrections aren’t made to account for significant exposure changes or other shifts in loss history
2. subject to sampling error since parameters are estimated from observed data