Venter Flashcards
What is the result if the Mack assumption hold?
Under Mack assumptions, the Chain Ladder method gives the minimum variance unbiased linear estimator of future claims emergence.
Explain how you can test Mack’s first assumption (4 testable implications)
- Significance of development factors (majority of abs(f(d)) are at least twice its std error, especially the earlier ones)
- Superiority of CL method to alternative emergence patterns
- Linearity of model (review residuals vs Loss: if non-linearity, this suggests emergence is a non-linear function of losses to-date)
- Stability of development factors (review residuals vs time: factors should not change over time).
If stable: use all Ays to calculate dev factors to reduce effects of random fluctuations and minimize variance.
If unstable: use wad average of factors with more weight to recent years.
Explain how you can test Mack’s second assumption (2 testable implications)
- No correlation among columns of development factors
- No particularly high/low diagonals (calendar year effects)
Identify 3 alternative emergence patterns
- Linear with a constant
E(Loss_d) = f(d)*Loss_k + g(d) - Factor times parameter (aka parameterized BF)
E(Loss_d) = f(d) * h(w) - Factor times parameter including calendar year effect
E(Loss_d) = f(d) * h(w) * g(w+d)
We can compare emergence patterns using which 3 goodness-of-fit tests?
- Adjusted SSE
- AIC
- BIC
Calculate Adjusted SSE
SSE = sum(E(Loss) - Loss)^2
E(Loss) = (LDF_d-1)Loss_d
Adjusted SSE = SSE/(n-p)^2
n = # incremental loss observations (excl. frist column)
p = # parameters in the model
p = 2#AYs - 2 if BF
p = #AYs - 1 if CL
Calculate AIC to test emergence patterns
SSE = sum(E(Loss) - Loss)^2
E(Loss) = (LDF_d-1)Loss_d
AIC = SSE * exp(2p/n)
n = # incremental loss observations (excl. frist column)
p = # parameters in the model
p = 2#AYs - 2 if BF
p = #AYs - 1 if CL
Calculate BIC to test emergence patterns
SSE = sum(E(Loss) - Loss)^2
E(Loss) = (LDF_d-1)Loss_d
BIC = SSE * exp(p/n)
n = # incremental loss observations (excl. frist column)
p = # parameters in the model
p = 2#AYs - 2 if BF
p = #AYs - 1 if CL
Name a disadvantage of AIC over BIC to test pattern emergence
AIC is too permissive of over-parameterization for large data sets. (advantage in fitting, but disadvantage in prediction)
Explain the Linear with Constant alternative emergence pattern.
E(Loss_d) = f(d) * Loss_k + g(d)
g(d) is often significant at very immature ages, especially highly variable slow repeating lines (ex: excess reinsurance).
If g(d) is significant, this emergence is more strongly supported than the chain ladder method.
For this pattern, p = 2*AYs - 2 (twice CL) due to inclusion of the g(d) parameters.
Explain the Cape Cod alternative emergence pattern.
E(Loss_d) = f(d) * h
Single h parameter for all accident years.
Requires a stable level of loss exposure over accident years, so we must adjust for exposure and price level differences among accident years.
For this pattern, p - #AYs - 1 (same as CL)
Cape Cod method works better for loss ratio triangles than for loss triangles, since a single target ultimate loss ratio makes more sense than a single target ultimate loss.
3 ways to reduce parameters in BF model
- Group Ays using apparent jump in loss levels and fitting a single h parameter to each group (can also group f parameters for ages with similar development factors)
- Cape Cod method: assume subsequent periods all have the same expected percentage development
- Use a trend line through the BF ultimate loss parameters to reduce accident year parameters to 2 instead of 1 for each year.
Calculate h(w) when constant variance (row factors)
h(w) = sumproduct(f(d);Loss_AY,d) / sum(f(d)^2)
Calculate f(d) when constant variance (column factors)
f(d) = sumproduct(h(AY;Loss_AY,d)) / sum(h(AY)^2)
Calculate E(Loss) under parameterized BF
E(Loss) = f(d) * h(AY)
Calculate h(w) when variance is proportional of f(d)h(w)
h(w)^2 = sum(Loss_AY,d^2 / f(d)) / sum(f(d))
Then, take square root to get h(w)!!!
Calculate f(d) when variance is proportional of f(d)h(w)
f(d)^2 = sum(Loss_AY,d^2 / h(AY)) / sum(h(AY))
Then, take square root to get f(d)!!!
Calculate h for Cape Cod method when constant variance
h(CC) = sumproduct(f(d), q(w,d)) / sum(f(d)^2)
Calclate h for Cape Cod method when variance is proportional to f(d)h(w)
h^2 = sum(q(w,d)^2 / f(d)) / sum(f(d))
Then, take the square root to get h!!!
2 ways to improve Cape Cod fit
- Use loss ratio triangle
- Adjust loss ratios for trend and rate level
What is the assumption of Chain Ladder for future loss emergence
Assumes future emergence is proportional to losses emerged to-date for a given accident year.
What is the assumption of BF for future loss emergence
Assumes expected emergence in each period id a % of ultimate loss.
Regards losses emerged to-date as a random component that does not influence future development.
If this is the case, using the CL will apply factors to the random component and increase error.
What is the assumption of the Cape Cod and additive chain ladder methods for future loss emergence.
Years with low (high) losses to-date have the same expected future dollar development as other accident years.
Explain how to test correlation of Development Factors
H0: factors are independent
r = coefficient correlation
T = r*sqrt((n-2)/(1-r^2))
n = # points in the column-pair
df = n-2
threhsold = t(df=n-2, a%)
If T > threshold, reject H0
Provide 2 methods that could improve age-to-age factors considering they are not stable over time and shows increasing trend.
- Use a 5y weighted average
- Adjust the data for systematic chages
Describe an alternative method of testing the stability of age-to-age factors based on venter factors (other than plotting factors vs accident year)
Plot raw residuals against time.
If there are strings of positives and negatives in a row, then development factors are not stable.
Discuss how the estimated future loss emergence for a year with abnormally high losses to date would differ under the chain ladder versus additive model
The future losses under chain-ladder method would be abnormally high since it would be applying a multiplicative factor to abnormally high losses to date.
Future losses under the additive model would not be impacted by abnormally high losses because the model is not impacted by losses to date.
