Venter Factors

Question 1

CL assumptions - incremental losses

Answer 1

1. future increm expected loss is prop to the cumulative losses to date
2. variance of the next increm loss is a fct of the age and the cumulative losses to date
3. losses for a given AY are independent of losses for any other AY

Question 2

Testable Implications of Assumptions

Answer 2

1. Significance of factor f(d)
2. Superiority of factor assumption to alternative emergence patterns
3. Linearity of model: look at residuals as a function of c(w, d)
4. Stability of factor: look at residuals as a function of time
5. No correlation among columns
6. No high or low diagonals

Question 3

testing significance of factors AKA testing implication 1

Answer 3

If factors from linear regression of increm losses against previous cumul loss are more than two times their standard deviations, then those factors are considered to be significantly different from 0

Question 4

alternative emergence patterns

Answer 4

linear with constant: E[q(w,d+1)]=f(d)c(w,d)+g(d)

factor times parameter: E[q(w,d+1)]=f(d)h(w)

including CY effects

Question 5

testing superiority of other emergence patterns AKA testing implication 2

Answer 5

to compare development development methods, should look at SSE

adjusted SSE: SSE/(n-p)^2 where n is predicted points

p = 2*AYs-2 for BF and AYs-1 for CL and CC

AIC: SSE*exp(2p/n)

BIC: SSE*n^(p/n)

Question 6

purpose of adjusted SSE

Answer 6

is intended to penalize methods that use too many parameters

more parameters gives adv in fitting but disadv in prediction

Question 7

testing significance of factors for linear with constant emergence pattern

Answer 7

if constant is significant and factor is not, additive development process may be indicated

Question 8

testing implications 1&2 graphically

Answer 8

plotting age d+1 losses against age d losses

*factor-only model (no constant) would show roughly a straight line through the origin with slope equal to the development factor

*A constant-only model (no factor) would show roughly a horizontal line at the height of the constant

Question 9

discussion of 1st assumption

Answer 9

If future loss emergence is a constant plus a percent of emergence to date, a factor plus constant development method should be used

If future loss emergence is proportional to ultimate losses rather than to emerged to date, a Bornhuetter/Ferguson approach is preferred

To test this assumption against its alternatives, we must fit the alternative methods to the data and apply a goodness-of-fit measure

Question 10

Iterative method for fitting a parameterized BF model when variance of residuals is constant over triangle

Answer 10

Use iterative method to minimize the sum of the squared residuals

Var prop to f(d)^p*h(w)^q where p=q=0

Question 11

Iterative method for fitting a parameterized BF model when variance of residuals is NOT constant over triangle

Answer 11

use weighted least squares if the variances of the residuals are not constant over the triangle

Var prop to f(d)^p*h(w)^q where p=q=1

Question 12

Iterative method for fitting a Cape Cod model

Answer 12

Same process as the BF model, except we have single h and formula is summed over w,d instead of d

Question 13

additive chain-ladder method and the Cape Cod method adjusted SSEs

Answer 13

they are the same value

Question 14

Chain-ladder method assumes

Answer 14

future emergence for an AY will be proportional to losses emerged to date

Question 15

BF method assumes

Answer 15

that expected emergence in each period will be a percentage of ultimate losses

Question 16

Cape Cod and additive chain-ladder methods assume

Answer 16

that years showing low losses or high losses to date will have the same expected future dollar development (because they assume a constant h over all accident years)

Question 17

testing implication 3 AKA linearity

Answer 17

scatter plot of the raw incremental residuals (actual emergence - expected emergence) against the previous cumulative losses provides a test for linearity

chain-ladder method assumes that the incremental losses are a linear function of the previous cumulative losses

If there are strings of positive and negative residuals in a row, then a non-linear process may be indicated

Question 18

testing implication 4 AKA stability of factors

2 tests

Answer 18

test1: Plot the incremental residuals against time

*If there are strings of positive and negative residuals in a row, then the development factors may not be stable

test2:Look at a moving average of a specific age-to-age factor

*If the moving average shows clear shifts over time, then instability exists and we may want to use a weighted average of the factors

*If the moving average shows large fluctuations around a fixed level, this does NOT mean we should focus only on recent data. In this case, a broader range of data is actually better

Question 19

testing implication 5 AKA correlation of development factors

Answer 19

Calculates the sample correlation coefficients for all pairs of columns in the development factor triangle, and then count how many of these are significant (at the 10% level) to determine if correlation exists

sample correlation: r = (X-EX)(Y-EY)/sqrt((X-EX)²(Y-EY)²)

r=(avg(xy)-avg(x)avg(y))/(stddev(x)*stddev(y))

stddev=sum(X-EX)^2/n

Test statistic T=r*sqrt((n-2)/(1-r²))

T is t-distributed with n-2 DOF where n is first n elements in both columns aka data points they share

Question 20

testing for # of significant correlations required @ 10% level to conclude correlation exists within triangle assuming tolerance of 2 std dev

Answer 20

M = (n-3) choose 2

column pairs that display sign corr ~Binomial RV(M,0.1)

M*0.1+2*sqrt(M*0.1*(1-0.9)

Question 21

testing implication 6 AKA Significantly high or low diagonals

Answer 21

• before run regression must create table from incremental triangle (5x5 example)
• create table: Increm loss ages 1-4, Cum0, Cum1, Cum2, Cum3, Dummy1, Dummy2, Dummy3
• Cum gets value from triangle if prior cumulative loss exits in that column of the triangle
• so dummy gets 1 if 1st column value in this table exists in that diag of triangle
• if absolute values of dummy coeff are < 2x their std dev, coeff are insignificant -> CY effects do not exist
• this test assumes applying standard CL

Question 22

under Mack assumptions, what does CL give us

Answer 22

In essence what Mack showed is that under assumptions, the chain ladder method gives the minimum variance unbiased linear estimator of future emergence