Brosius

Question 1

Link Ratio Method

Answer 1

Y is estimated as L(x)=cx, where c is a ‘selected link ratio’

c is a weighted average of several years

Question 2

Budgeted Loss Method

Answer 2

Used when fluctuation in loss experience is extreme or past data is not available

a value k is chosen so that L(x)=k

k can either be an average of y over several years, or multiplying earned premium by an expected loss ratio

Question 3

Least Squares Method

Answer 3

• Estimates L(x) by fitting a line to the points (x,y)
• L(x) = a +bx, where a and be are determined by the least squares fit:
  
  • b= (mean(xy) -mean(x)mean(y))/(mean(x²)-mean(x)²)
  • a=mean(y)-b*mean(x)
• Important advantage of the least squares method is it’s flexibility. Gives more or less weight to the observed value of x as appropriate.

Question 4

Special Cases of Least Squares Method

Answer 4

• Link Ratio Method when (a=0)
• Budgeted Loss Method when (b=0)
• Bornhuetter Ferguson Method when (b=1)

Question 5

Parameter Estimation Errors

Answer 5

• Significant changes in the nature of loss experience and sampling error can lead to values of a and b that don’t reflect reality
• When a<0, y will be negative for small values of x (use link ratio method)
• When b<0, y decreases as x increases (use budgeted loss method)

Question 6

Simple Model for Least Squares Method

Answer 6

• Q(x) = E[Y|X=x]
• R(x) = E{Y-X|X=x] = Q(x)-x
• Q(x) represents the total number of claims
• R(x) represents the expected number of claims outstanding

Question 7

General Poisson-Binomial case

Answer 7

• Y is a poisson distributed with mean µ, and any given claim has probability d of being reported by year-end
• Q(x) = x + µ(1-d)
• R(x) = µ(1-d)
• Bornhuetter/Ferguson is optimal in this case because it doesnt not depend on the number of claims already reported

Question 8

Negative Binomial-Binomial case

Answer 8

• Y is a negative binomial distributed with parameters (r,p), and where any given claim has probability d of being reported by year end
• R(x) = [(1-d)*(1-p)]/[1-(1-d)(1-p)] * (x+r)
• Except when d=1, this is an increasing linear function of x
• Increase in reported claims leads to an increase in estimate of outstanding claims (No Budgeted Loss or BF Methods)
• Relationship is ont proportional, so link ratio method isn’t optimal

Question 9

Fixed Prior Case

Answer 9

• Y is not random; there is a value k such that Y is sure to equal k
• Q(x)=k and R(x) = k-x
• Budgeted Loss Method

Question 10

Fixed Reporting Case

Answer 10

• Suppose there is a number d≠0 such that the percentage of claims reported by year-end is always d
• Q(x) = x/d and R(x) = x/d - x
• Link Ratio Method

Question 11

Linear Approximation for Bayesian Credibility (Advantages)

Answer 11

• As a replacement for Bayesian estimate linear approximation has advantages:
  
  • Simpler to compute
  • Easier to understand and explain
  • Less dependent upon the underlying distribution

Question 12

Relationship of Cov(X,Y) and Var(X)

Answer 12

• Cov(X,Y)<var : budgeted loss method>
  <li>Cov(X,Y)=Var(X) : Bornheutter Ferguson Method</li><li>Cov(X,Y)&gt;Var(X) : Chain Ladder Method</li></var>

Question 13

Least Squares Method Appropriateness

Answer 13

• Does not make sense if year to year changes in loss experience are largely due to systematic shifts or distortions in book of business
• May be appropriate if year to year changes are due to random chance
• If there are systematic distortions you can make adjustments and still use least squares
  
  • Correcting for inflation
  • Business expansion->dividing by an exposure base

Question 14

Development Formula 1

Answer 14

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

Question 15

Variance of Hypothetical Means

Answer 15

• VHM = Var_y(E[X|Y])
• Represents the variability resulting from the loss occurrence process

Question 16

Expected Value of the Process Variance

Answer 16

• E_y[Var(X|Y)]
• EVPV represents the variability resulting from the loss reporting process

Question 17

Development Formula 2

Answer 17

• L(x) = Z(x/d) + (1-Z)E[Y]
• Z = VHM/(VHM + EVPV)
• If EVPV = 0, we give full weight to link ratio estimate
• If VHM = 0, we give full weight to budgeted loss estimate

Question 18

Caseload Effect

Answer 18

• Development Formula 2 assumes that the expected number of claims reported is proportional to the number of claims incurred
• Since a claim is more likely to be reported quickly when caseload is low, we expect development ratio to be a decreasing function of y, not a constant
• If E[X|Y=y] = dy + x₀ we get a development ratio of d + x₀/y which decreases as y increases
• When x₀=0 we obtain Development Formula 2 as a special case

Question 19

Development Formula 3

Answer 19

• L(x) = Z(x-x₀)/d + (1-Z)E[Y]
• Z = VHM(VHM+EPV)