Clark

Question 1

Briefly describe the 2 main objectives for a statistical loss reserving model.

Answer 1

1. Have a tool to describe loss emergence mathematically that can aid in selecting carried reserves.
   Expected loss emergence
2. Have a model that estimates a range around the expected reserve.
   Distribution of loss emergence around the expectation

Question 2

State the 2 underlying causes for reserve variability

Answer 2

1. Process variance: uncertainty due to randomness
2. Parameter variance: uncertainty in the estimate of expected value, estimation error

Question 3

Briefly describe the expected loss emergence pattern

Answer 3

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.

Question 4

Calculate the cumulative % paid using the loglogistic loss emergence pattern

Answer 4

G(x) = x^w / (x^w + theta^w)

Question 5

Calculate the cumulative % paid using the Weibull loss emergence pattern

Answer 5

G(x) = 1 - exp(-x/theta)^w

Question 6

Briefly compare the Weibull and loglogistinc loss emergence curves

Answer 6

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)

Question 7

When will a Weibull or Loglogistic claim emergence model not work?

Answer 7

When there is real, expected negative development (e.g. salvage and subrogation)

This model still works if some data points show negative development

Question 8

State 3 advantages of using parametrized curves to describe expected loss emergence patterns

Answer 8

1. Estimation is simple since we only have to estimate two parameters
2. Can use data that is not from a triangle with evenly spaced evaluation data
3. Final pattern is smooth and does not follow random movements in the historical age-to-age factors.

Question 9

Briefly describe the underlying assumption of the LDF method

Answer 9

Assumes ultimate loss in each accident year is independent of losses in other accident years

Question 10

Briefly describe the underlying assumption of the Cape Cod method

Answer 10

Assumes a constant expected loss ratio across all accident years

Question 11

Calculate the expected incremental loss emergence for the LDF method

Answer 11

mu = ULT_AY*(G(x_k) - G(x_k-1))

E(IncLoss) = Ult_AY * % incremental emergence

Question 12

Calculate truncated LDF

Answer 12

LDF_trunc = G(x_trunc) / G(x)

If no truncation:
LDF = 1/G(x)

Question 13

Calculate the expected incremental loss emergence for the Cape Cod method

Answer 13

mu = Prem * ELR * (G(x_k) - G(x_k-1))

E(IncLoss) = Expected Loss * % Incremental Emergence

Question 14

Which expected loss emergence method is preferred according to Clark and why?

Answer 14

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)

Question 15

How does the Cape Cod method take advantage of more information?

Answer 15

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.

Question 16

Calculate the Varian/Mean Ratio

Answer 16

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

Question 17

Calculate the MLE for estimating best-fit parameters

Answer 17

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

Question 18

Calculate the Coefficient of Variation of Reserves

Answer 18

Std Dev(Rsv) = (ProcessVar + ParameterVar)^0.5

CV = StdDev(Rsv) / Rsv

coeff of var = sd / mean

Question 19

Calculate the process variance of Reserves

Answer 19

ProcessVar = s^2 * Reserve

Question 20

State the 3 key assumptions in Clark’s model

Answer 20

1. Incremental losses are independent and iid
2. The variance/mean scale parameter s^2 is fixed and known
3. Variance estimates are based on an approx to the Rao-Cramer lower bound

Question 21

Briefly describe the key assumption of LDF Curve Fitting and why they may not hold in practice (assumption 1)

Answer 21

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

Question 22

Briefly describe the second key assumption of LDF curve fitting

Answer 22

The model ignore the variance on the variance

Question 23

Briefly describe the impact of the key assumptions about the LDF curve fitting model

Answer 23

Future loss emergence potentially may have more variability than what the model produces.

Question 24

Calculate normalized residual

Answer 24

r = (IncLoss - mu) / (s^2 * mu)^0.5

norm residual = actual incremental - expected incremental / (s^2 * expected incremental)^0.5

Question 25

Briefly explain how to test if a fixed s^2 is an appropriate assumption

Answer 25

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

Question 26

Briefly describe the exposure bases for the Cape Cod method

Answer 26

1. 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.
2. Original loss projections by year
3. Estimated Claim Counts

Question 27

Calculate the Variance of Prospective Losses

Answer 27

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

Question 28

Calculate the Variance of CY Development using the LDF method

Answer 28

Est Dev = Ult * (G(x+12)-G(x))

ProcessVar_CYDev = s^2 * sum of Est Dev

Question 29

Calculate the variance of CY Development using the Cape Cod method

Answer 29

Est Dev = Prem * ELR * (G(x+12)-G(x))

ProcessVar_CYDev = s^2 * sum of Est Dev