Clark Flashcards
Briefly describe the 2 main objectives for a statistical loss reserving model.
- Have a tool to describe loss emergence mathematically that can aid in selecting carried reserves.
Expected loss emergence - Have a model that estimates a range around the expected reserve.
Distribution of loss emergence around the expectation
State the 2 underlying causes for reserve variability
- Process variance: uncertainty due to randomness
- Parameter variance: uncertainty in the estimate of expected value, estimation error
Briefly describe the expected loss emergence pattern
G(x) = 1/LDF(x)
G(x) is the cumulative % of loss paid (or reported) as of time x
x represents the time in months from avg accident date to the eval date.
Calculate the cumulative % paid using the loglogistic loss emergence pattern
G(x) = x^w / (x^w + theta^w)
Calculate the cumulative % paid using the Weibull loss emergence pattern
G(x) = 1 - exp(-x/theta)^w
Briefly compare the Weibull and loglogistinc loss emergence curves
Weibull generally results in a smaller tail factor (thinner tail)
With the loglogistinc curve, there is more extrapolation in the tail (might consider using a truncation point)
When will a Weibull or Loglogistic claim emergence model not work?
When there is real, expected negative development (e.g. salvage and subrogation)
This model still works if some data points show negative development
State 3 advantages of using parametrized curves to describe expected loss emergence patterns
- Estimation is simple since we only have to estimate two parameters
- Can use data that is not from a triangle with evenly spaced evaluation data
- Final pattern is smooth and does not follow random movements in the historical age-to-age factors.
Briefly describe the underlying assumption of the LDF method
Assumes ultimate loss in each accident year is independent of losses in other accident years
Briefly describe the underlying assumption of the Cape Cod method
Assumes a constant expected loss ratio across all accident years
Calculate the expected incremental loss emergence for the LDF method
mu = ULT_AY*(G(x_k) - G(x_k-1))
E(IncLoss) = Ult_AY * % incremental emergence
Calculate truncated LDF
LDF_trunc = G(x_trunc) / G(x)
If no truncation:
LDF = 1/G(x)
Calculate the expected incremental loss emergence for the Cape Cod method
mu = Prem * ELR * (G(x_k) - G(x_k-1))
E(IncLoss) = Expected Loss * % Incremental Emergence
Which expected loss emergence method is preferred according to Clark and why?
The Cape Cod method is preferred.
When using a development triangle, data is summarized into relatively few data points for a model.
This results in the problem of overfitting with the LDF method, which has n+2 parameters to fit.
The Cape Cod only has 3 parameters.
The Cape Cod method uses more information (premium exposure)
How does the Cape Cod method take advantage of more information?
Cape Cod uses exposure base.
This may lead to somewhat higher process variance, but usually results in much smaller estimation error.
Key point:
Additional information reduces the variance in the reserve analysis, which also produces a better reserve estimate.
Calculate the Variance/Mean Ratio
s^2 = 1/(n-p) * sum(IncLoss - mu)^2 / mu
s^2 = 1/(n-p) * sum(actual - expected)^2 / expected
n = # cells in loss triangle
p = # parameters
Calculate the MLE for estimating best-fit parameters
Over-dispersed Poisson distribution:
likelihood = product(lambda^(c/s^2) * exp(-lambda))/(c/s^2)!
lambda = mu/s^2
Loglihood = l = sum of (IncLoss * ln(mu) - mu) = sum MLE term
Calculate the Coefficient of Variation of Reserves
Std Dev(Rsv) = (ProcessVar + ParameterVar)^0.5
CV = StdDev(Rsv) / Rsv
coeff of var = sd / mean
Calculate the process variance of Reserves
ProcessVar = s^2 * Reserve
State the 3 key assumptions in Clark’s model
- Incremental losses are independent and iid
- The variance/mean scale parameter s^2 is fixed and known
- Variance estimates are based on an approx to the Rao-Cramer lower bound
Briefly describe the key assumption of LDF Curve Fitting and why they may not hold in practice (assumption 1)
Independence
One period does not affect surrounding period, but:
There may be positive correlation due to inflation on all periods
Neg corr if a large settlement replaces later payment streams
Identically Distributed
Emergence patterns are the same for all accident years, but:
Different risks and business mix would have been written in different accident years with different claims handling processes
Briefly describe the second key assumption of LDF curve fitting
The model ignore the variance on the variance
Briefly describe the impact of the key assumptions about the LDF curve fitting model
Future loss emergence potentially may have more variability than what the model produces.
Calculate normalized residual
r = (IncLoss - mu) / (s^2 * mu)^0.5
norm residual = actual incremental - expected incremental / (s^2 * expected incremental)^0.5
Briefly explain how to test if a fixed s^2 is an appropriate assumption
Plot the residuals vs expected incremental loss to check for a random distribution around zero
s^2 is not constant if residuals are more clustered around zero at either high or low expected incremental loss
Briefly describe the exposure bases for the Cape Cod method
- On-Level Premium
Premium adjusted to a common rate level per exposure
Better than unadjusted so that market cycles do not distort results
Cape Cod method assumes a constant ELR across accident years.
We can make additional adjustment for less trend net of exposure trend to get all years on the same cost level. - Original loss projections by year
- Estimated Claim Counts
Calculate the Variance of Prospective Losses
E(Loss_prosp) = Prem * ELR
Process Var = s^2 * E(Loss_prosp)
Paramter Var = Var(ELR) * Prem^2
SD(Loss_prosp) = (ProcessVar + ParameterVar)^0.5
Calculate the Variance of CY Development using the LDF method
Est Dev = Ult * (G(x+12)-G(x))
ProcessVar_CYDev = s^2 * sum of Est Dev
Calculate the variance of CY Development using the Cape Cod method
Est Dev = Prem * ELR * (G(x+12)-G(x))
ProcessVar_CYDev = s^2 * sum of Est Dev
Identify the direction in which the coeff of variation of reserver would change if method used were changed from LDF to Cape Cod.
CV will decrease because Cape Cod method uses more information (on-level premium), thus results in better reserve estimate, lower std dev and lower CV.
Explain why s^2 for the LDF method is higher than s^2 for the Cape Cod method
s^2 refers to process variance.
When calculating s^2, we divide by (n-p), where p is the number of parameters. Since the LDF method requires more parameters, it has higher s^2.
Briefly explain why the Weibull growth curve is more appropriate for a ST LOB
Because it has higher tail, thus terminates sooner than the loglogistic curve.
Identify an approach to reduce parameter variance in a statistical model
Reduce the number of parameters in the model will reduce parameter variance.
Briefly explain when a curve-fitting method for selecting loss emergence patterns will produce a higher mean estimate of ultimate losses than a weighted average method.
Curves naturally create a tail factor by going from 0% to 100% emergence whereas weighted average methods cannot produce factors past the triangle where no data exists. Tail factor produces a higher mean estimate for curve-fitting method.
Identify one reason why curve-fitting might be better than weighted average for estimating payment pattern.
Provide estimates of development after the end of available data.
Identify one reason why weighted-average development factors might be better than curve-fitting for estimating payment pattern.
Simpler to calculate.
Briefly explain why loss development method using weighted average produces lower standard deviation of ult losses than curve fitting.
Weighted average loss development factor ignores volatility in the tail which reduces variability of ultimate losses.
State an advantage of maximizing log-likelihood function to get MLE parameters.
It works in the presence of negative or zero incremental losses since we never actually take the log of cumulative losses.
