Brosius

Question 1

L(x)

(Least Squares estimate)

Answer 1

L(x) = a + bx

where

b= (avg(xy) - avg(x)avg(y)) / (avg(x²) - avg(x)²)

and

a = avg(y) - b * avg(x)

Question 2

Parameter Estimation errors

Answer 2

Signifcant changes in nature of loss experience and sampling error can lead to values of a and b that do not reflect reality

When a<0, our estimate of y will be negative for small values of x (use link ratio method)

When b<0, our estimate of y decreases as x increases (use budgeted loss method)

Question 3

Q(x)

Answer 3

total number of claims

E[Y| X = x]

Question 4

R(x)

Answer 4

expected number of claims outstanding

E[Y-X|X=x] = Q(x) - x

Question 5

General Poisson-binomial case

u= mean

d = probability of any given claim being reported by year end

Answer 5

Q(x) = x + u(1-d)

R(x) = u(1-d)

R(x) does not depend upon number of claims already reported, so B-F estimate is optimal

Question 6

Negative Binomial- Binomial case

with parameters (r,p) and d

Answer 6

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

increasing linear function of x, budgeted loss, B-F, and link ratio method not optimal

Question 7

Fixed Prior Case

Answer 7

Y is not random; there is some value k such that Y is sure to equal k

Q(x) = k

R(x) = k-x

corresponds to budgeted loss method

Question 8

Fixed reporting case

Answer 8

there is a number d such that the percentage of claims reported by year end is always d

Q(x) = x/d

R(x) = x/d - x

Corresponds to link ratio method

Question 9

Advantages of best linear approximation of Q over Bayesian estimate

Answer 9

difficult to compute a pure Bayesian estimate since it requires knowledge of the loss and loss reporting processes

best linear approximation is:

• simpler to compute
• easier to understand and explain
• less dependent upon the underlying distribution

Question 10

Best linear approximation to Q (L(x))

Answer 10

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

Question 11

Cov(X,Y)<var>
</var>

Answer 11

a large reported amount should lead to a decrease in the reserve (budgeted loss method)

Question 12

Cov(X,Y) = Var(X)

Answer 12

a large reported amount should not affect the reserve (B-F Method)

Question 13

Cov(X,Y) > Var(X)

Answer 13

a large reported amount should lead to an increase in the reserve (link ratio method)

Question 14

When least squares development is appropriate

Answer 14

• does not make sense if year to year changes in loss experience are due largely to systematic shifts or distortions in the book of business
• the least squares fit may be appropriate if year to year changes are due largely to random chance

Question 15

Data adjustments before applying least squares method

Answer 15

• Incurred loss data - correct for inflation by putting the years on a constant-dollar basis before fitting a line
• If the business expands, divide each year’s losses by an exposure base to eliminate distortion