Link Ratio Method
Y is estimated as L(x)=cx, where c is a ‘selected link ratio’
c is a weighted average of several years
Budgeted Loss Method
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
Least Squares Method
- 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.
Special Cases of Least Squares Method
- Link Ratio Method when (a=0)
- Budgeted Loss Method when (b=0)
- Bornhuetter Ferguson Method when (b=1)
Parameter Estimation Errors
- 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)
Simple Model for Least Squares Method
- 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
General Poisson-Binomial case
- 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
Negative Binomial-Binomial case
- 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
Fixed Prior Case
- 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
Fixed Reporting Case
- 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
Linear Approximation for Bayesian Credibility (Advantages)
- As a replacement for Bayesian estimate linear approximation has advantages:
- Simpler to compute
- Easier to understand and explain
- Less dependent upon the underlying distribution
Relationship of Cov(X,Y) and Var(X)
- Cov(X,Y)<var : budgeted loss method>
<li>Cov(X,Y)=Var(X) : Bornheutter Ferguson Method</li><li>Cov(X,Y)>Var(X) : Chain Ladder Method</li></var>
Least Squares Method Appropriateness
- 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
Development Formula 1
L(x) = (x-E[x])*Cov(X,Y)/Var(X) + E[Y]
Variance of Hypothetical Means
- VHM = Vary(E[X|Y])
- Represents the variability resulting from the loss occurrence process
Expected Value of the Process Variance
- Ey[Var(X|Y)]
- EVPV represents the variability resulting from the loss reporting process
Development Formula 2
- 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
Caseload Effect
- 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
Development Formula 3
- L(x) = Z(x-x0)/d + (1-Z)E[Y]
- Z = VHM(VHM+EPV)
