Brosius

Question 1

3 Special Cases for Brosius Least Squares

Answer 1

Chain Ladder (a = 0)
BF (b = 1)
Budgeted Loss (b = 0)

Question 2

Claim counts Poisson and d = probability of reporting in first period, what is the Bayes reserve?

Answer 2

R(x) = mu(1 -d)

Question 3

Negative binomial distribution

Answer 3

E[Y] = r(1-p)/p

Question 4

Claim counts NegBin(r,p), d is probability of reporting in first period, x is actual reported, what is the Bayes reserve?

Answer 4

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

Question 5

Formulas for a and b in Brosius Least Squares

Answer 5

b = [mean(xy) - mean(x)mean(y)]/[mean(x^2)- mean(x)^2]

Question 6

For ultimate Y and Reported loss X, what is best linear estimate of Y given X?

Answer 6

L(x) = x - E[x] + E[Y]

Question 7

When is Brosius appropriate/inappropriate

Answer 7

Appropriate: when distribution is same across multiple years
Inappropriate: year to year changes are systematic shifts (inflation, legal environment)

Question 8

Estimate L(x) using credibility

Answer 8

L = Z(x/d) + (1-Z)(E[Y])
Z = VHM/(VHM + EVPV)

Question 9

Statistics for Brosius

Answer 9

c = mean(y)/mean(x)
Z = b/c
d = 1/c