Brosius

Question 1

When should you use the least squares method

Answer 1

Whenever random year-to-year fluctuations in loss experience are significant - aka changes are largely due to random chance

Does not make sense if there are systematic shifts or distortions to the book of business

Question 2

Link Ratio Method

Answer 2

L(x) = cx
where L(x) is the estimate of y (losses at future eval) and c is the volume weighted average CDF

fits a line through the origin with a slope equal to the selected link ratio

Question 3

Budgeted Loss Method

Answer 3

L(x) = k
where k is a constant
Used when there are extreme fluctuations in losses or past data is not available
Brosisus assumes k is chosen by averaging y over several years

fits a horizontal line equal to the average of y

Question 4

Least Squares Method

Answer 4

L(x) = a + bx
Estimated y by fitting a line to the points (x,y) that minimizes the sum of the squares of the residuals
Use LINEST(y,x) in Excel -> output is (b,a)

Question 5

Special Cases of LS

Answer 5

a = 0 -> Link Ratio Method
b = 0 -> Budgeted Loss Method
b = 1 -> BF Method

Question 6

Parameter Estimation Errors

Answer 6

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

• when a < 0, y is negative for small x and the Link Ratio method should be used
• when b < 0, y decreases as x increases and the Budgeted Loss method should be used

Question 7

Hugh White’s Question

Answer 7

Question: You are trying to establish the reserve for commercial auto bodily injury and the reported proportion of expected losses as of the statement date for the current accident year period is 8% higher than it should be. Do you:

• Reduce the bulk reserve by a corresponding amount (Budgeted Loss method)
• Leave the bulk reserve at the same percentage level of expected losses (BF method)
• Increase the bulk reserve in proportion to the increase of actual reported over expected reported (Link Ratio method)

Question 8

Systemic distortions

Answer 8

If the data can be adjusted, we can use the LS method:

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

Question 9

Development Formula 1

Answer 9

Given random variables 𝑌 describing ultimate losses and 𝑋 describing reported losses, the best linear approximation to 𝑄 (pure Bayesian estimate) is as follows:

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

• If Cov(X,Y) < Var(X), a large reported amount should lead to a decrease in the reserve (Budgeted Loss method)
• If Cov(X,Y) = Var(X), a large reported amount should not affect the reserve (BF method)
• If Cov(X,Y) > Var(X), a large reported amount should lead to an increase in the reserve (Link Ratio method)

Question 10

Expected Value of the Process Variance (EVPV)

Answer 10

E_Y[Var(X|Y)] = Var(X/Y) * (Var(Y) + E(Y)^2)

Represents the variability resulting from the loss reporting process

Question 11

Variance of the Hypothetical Mean (VHM)

Answer 11

Var_Y[E(X|Y)] = E(X/Y)^2 * Var(Y)

Represents the variability resulting from the loss occurrence process

Question 12

Development Formula 2

Answer 12

L(x) = Z * x/d + (1 - Z) * E(Y)

where Z = VHM / (VHM + EVPV)

L is a credibility weighting of Link Ratio estimate x/d and Budgeted Loss estimate E(Y)

Question 13

LS Credibility Weighted Estimate

Answer 13

L(x) = Z * x/d + (1 - Z) * E(Y)

Z = bd = b/c

where c is the volume-weighted average CDF

If using LRs, c = Avg Ult LR / Avg Undeveloped LR

Best linear approximation to a pure Bayesian approach, so it has the optimal credibility weighting

Question 14

LS Algorithm

Answer 14

Given cumulative reported/paid losses and EP:

1. Divide the losses by EP to adjust for systematic changes if EP is increasing or decreasing over time
2. Calculate ultimate LRs for years that are closest to ultimate using given tail factor
3. Calculate the LS parameters and ensure there are no parameter estimation errors
4. Calculate the next year’s ultimate LR using the LS parameters
5. Re-run the LS parameter estimation including the previous year’s estimated ultimate LR

Question 15

Caseload Effect

Answer 15

Since a claim is more likely to be reported quickly when the caseload is low, we expect the development ratio E(X|y)/y to be a decreasing function of y, not a constant

E(X|y) = dy + x_0

L(x) = Z * (x - x_0)/d + (1 - Z) * E(Y)
where Z = VHM / (VHM + EVPV)

Question 16

Caseload Effect Algorithm

Answer 16

1. Set up system of equations to solve for d and x_0
2. Use formula to estimate the ultimate losses L(x)

Question 17

Advantages of LS Method

Answer 17

• the ability to flex to other methods
• easy to implement and uses easily accessible data
• works well for developing losses for small states or lines that are subject to serious fluctuations

Question 18

Disadvantages of LS Method

Answer 18

• can lead you astray if corrections are not made to account for significant exposure changes or other shifts in loss history
• is subject to sampling error since parameters are estimated from observed data

Question 19

When is the Bayesian method appropriate?

Answer 19

• new LOB
• significant changes to the book of business and the go-forward experience will be different from historical

Question 20

Advantages of using the best linear approximation as a replacement for the pure Bayesian estimate

Answer 20

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