Correlation of age-to-age factors
Venter takes the columns of age-to-age factors and subtracts 1 from each value; correlation is the same with or without this step
r = CORREL(array1, array2)
T = r * sqrt( (n-2)/(1-r^2) ) with n-2 degrees of freedom
n is the # of LDF pairs being compared between the two columns
Two-sided test, take the absolute value of T to compare to the threshold or =T.inv() to get the probability for one side
Correlation of age-to-age factors for the entire triangle
Calculate every possible pair of columns’ correlation, T stats, and significance level;
Benchmark = sig level% * (#pairs tested) + sqrt(#pairs tested)
If # significant pairs > benchmark then significant correlation exists in the triangle
Parameterized BF Model after one iteration of h(w) and f(d)
Get cumulative paid triangle (if only given incremental) and calculate volume-weighted LDF and CDF to start
initial f(d) = incremental %paid to start but they drift away from that meaning after the iteration process; 1/CDF - prev f(d)s
First Iteration:
row factors h(w) = SUM across AY( f(d) * Inc Loss) / SUM (f(d)^2)
column factors f(1) = SUM( h(w) * Inc Loss) / SUM( h(w)^2)
-> these likely will not sum to 1.0
Calculate expected future incremental losses
E[q(w,d)] = f(d) * h(w)
Square the incremental triangle and calculate reserve or ultimate, whatever the problem asks you for.
Parameterized BF, constant variance, Cape Cod version
Start with f(d) as % incremental paid from chainladder method as usual;
Initial h(w) is only one value, calculated for the entire triangle (rather than BF for each AY):
hCC(w) =SUM across triangle (f(d) * Inc Loss) / SUM(f(d)^2)
Parameterized BF: Variance proportional to f(d)h(w)
Starting values of f(d)
First iteration:
h(w)^2 = SUM across row( IncLoss^2 / f(d) ) / SUM(f(d))
h(w) = SQRT(h(w)^2)
f(d)^2 = SUM down column( IncLoss^2 / h(w)) / SUM(h(w))
f(d) = SQRT(f(d)^2)
Calc expected future incremental losses:
E[Inc Loss] = f(d) * h(w)
Test for Significantly High/Low Diagonals
Need both Incremental and Cumulative triangles; generally ignore age 0 incrementals because there are no prior cumulatives
Table uses incremental losses ages 1+ as the dependent variable and prior cumulative losses as the independent variables
Dummy variables for diagonals numbered 2+ because Diagonal 1 is the baseline for the regression.
Inc Loss | CumLoss age 0 (only filled out for inc loss at age 1) | CumLoss age 1 (for IncLoss age 2) | etc. | Dummy 1 (Diag 2) | Dummy 2 (Diag 3)
Model output gives coefficients for each variable and std. dev.
If |Coef| / Std.Dev > 2 then it is considered significantly high or low
Additive chain ladder table format
Dummy variables instead of cumulative loss values
IncLoss (values) | Age 0 | Age 1 | Age 2 (all dummy 1s)
Goodness of Fit Test (SSE and Adj SSE)
SSE = Sum (Act - Exp)^2
Adj SSE = SSE / (n-p)^2
n = #inc loss obs
p = #parameters
Lowest SSE, AIC, or BIC is best
AIC
SSE * exp(2p/n)
BIC
SSE * n^(p/n)
BF # parameters
2 * #AYs - 2 (twice CL)
CL # parameters
AYs - 1
CC # parameters
AYs - 1 (same as CL)
Mack Assumptions Restated for Venter
- Expected incremental loss emergence is proportional to cumulative losses to date
- Losses are independent between AYs
- Variance of the next incremental loss is a function of age and cumulative losses to date
Testable Implication 1
Assumption 1 - Significance of development factors
If inc losses are proportional to losses-to-date then the dev factors SHOULD be significant. Test with regression (could include a constant, but more direct without one)
Testable Implication 2
Assumption 1 - Superiority to alternative loss emergence patterns
Goodness of fit tests on Linear, Linear with a constant, Parameterized BF, etc. to see if Linear really is the best predictor of incremental emerging losses
Testable Implication 3
Assumption 1 - Linearity of the model: Residuals vs Lossk
If residuals show a pattern then the incremental loss emergence triangle is a non-linear function of losses-to-date
Testable Implication 4
Assumption 1 - Stability of Development Factors: Residuals vs Time
Patterns of high/low residuals imply that the same development factors are not appropriate for all AYs and thus invalidate Ass1
Testable Implication 5
Assumption 2 - Correlation of dev factors
Correlation is evidence against CL method
* Conflict with Mack - he says this would be Ass1 while Venter says Ass2
Testable Implication 6
Assumption 2 - Significantly high or low diagonals (CY effects)
Run a regression of incremental losses against cumulative losses at prior dev periods and include a dummy var for each diagonal. If any dummies are statistically significant then CY effects are indicated
