Meyers

Question 1

State 3 reasons why model may not accurately predict distribution of actual outcomes.

Answer 1

1. Insurance losses are too dynamic to be fully reflected in a single stochastic model.
2. Other models may better fit the data.
3. Data used for calibrating the model might be missing key information to make accurate predictions.

Question 2

Name 2 graphical tests for validity of Mack Model assumptions (uniform distribution of percentiles)

Answer 2

1. Histogram: bars of equal height
2. P-P plot: sorted predicted percentiles should follow the expected percentiles along 45 degrees line
   expected = 100*i/(n+1)

Question 3

Briefly explain how to test Model Validity using Kolmogorov-Smirnov Test

Answer 3

Critical value: 136/n^0.5
n = # percentiles
expected = fi = 100*i/n
Di = abs(pi - fi)
D = max(Di)

H0: percentiles are uniformly distributed
If D > critical value, reject H0
If D < critical value, cannot reject H0 (conclude that percentiles are uniformly distributed and Mack Model is appropriate)

Question 4

How can you recognize a Ligh-Tailed model from histogram and p-p plot

Answer 4

Tallest bars at lowest and highest percentiles in histogram.

P-P plot has an S-shape.

Question 5

How can you recognize a Heavy-Tailed model from histogram and p-p plot

Answer 5

Tallest bars in the middle of histogram (normal distribution shape)

P-P plot has an inverted S-shape

Question 6

How can you recognize a Biased high model from histogram and p-p plot

Answer 6

Tallest bar at lowest percentile

P-P plot is bowed down.

Question 7

How can you recognize a valid model from histogram and p-p plot

Answer 7

P-P plot follows a straight 45 degrees line

Bars in histogram are of equal height.

Question 8

Is Mack Model for Incurred Loss validated?

Answer 8

No, Incurred Mack model is too light in the tails.

Histogram shows more high/low percentiles than expected

P-P plot shows a slanted S curve and K-S test fails at the 5% level for all lines combined.

Question 9

Is Mack Model for Paid Loss validated?

Answer 9

No, Paid Mack model is biased high.

Histogram and p-p plot show more low percentiles than expected.

K-S test fails at 5% for all lines combined.

Question 10

Is Bootstrap ODP Model for Paid Loss validated?

Answer 10

No, ODP Bootstrap model is biased high.

Histogram and p-p plot show more low percentiles than expected.

K-S test fails at the 5% level for all lines combined.

Question 11

Briefly explain Leveled Chain Ladder (LCL) model and whether it is validated

Answer 11

LCL uses random level parameters for each AY.
uw,d = aw + bd
Loss_w,d follow lognormal distribution (uw,d , sigma_d)

LCL is still too light in the tills but better than Mack.

Histogram and p-p plot show more high/low percentiles than expected.

K-S test fails at 5% for all lines combined.

Question 12

Briefly explain the Correlated Chain Ladder (CCL) model and whether it is validated

Answer 12

CCL allows for correlation between accident years.
u1,d = a1 + bd
uw,d = aw + bd + rho*(ln(C_w-1,d) - u_w-1,d)
Loss_w,d follow lognormal distribution (uw,d , sigma_d)

Rho is generally positive (result in higher prediction variance than LCL)

CCL is validated, but has mildly thin tails.

Histogram and p-p plot show slightly more high/low percentiles than expected and the K-S PASSES the 5% level.

Question 13

Briefly explain the Correlated Incremental Trend (CIT) model and whether it is validated

Answer 13

CIT includes a calendar-year trend for incremental paid loss.

CIT allows for correlation between Ays (like CCL)

CIT is biased high on paid losses.

Histogram and p-p plot show more low percentiles than expected.

K-S test fails at the 5% level for all lines combined.

Question 14

Calculate Inc Loss from CIT model

Answer 14

uw,d = aw + bd + tau(w+d-1)
Zw,d follows lognormal distribution (uw,d , sigma_d)
Inc Loss_1,d follows normal (Z1,d , delta)
Inc Loss_w,d follows normal (Zw,d + rho(IncLoss_w-1,d - Zw-1,d)*e^tau , delta)

LIT is a special case with rho=0.

Question 15

Briefly explain the Changing Settlement Rates (CSR) model and whether it is validated.

Answer 15

CSR parameter v reflects changes to the claims settlement rate.

uw,d = aw + bd*(1-v)^w-1
Loss_w,d follows lognormal (uw,d , sigma_d)

v is generally positive, reflecting a speedup in payment pattern.

Histogram and p-p plot show that CSR corrects the high bias of other Bayesian paid models and K-S test passes at 5%.

CSR model is validated.

Question 16

Define Process Risk

Answer 16

Average variance of outcomes from expected result.

Question 17

Define Parameter Risk

Answer 17

Variance due to uncertainty in the parameters, reflected in the posterior distributions of the parameters.

Represents the overwhelming majority of the total risk in the loss data sets that Meyers looks at.

Question 18

Define Model Risk

Answer 18

The risk that we did not select the “correct” model.

Meyers does not explore model risk. It could be reflected by weighting together multiple models.

Question 19

Define Total Risk

Answer 19

Total Risk = Process Risk + Parmeter Risk
Total Risk = EVPV + VHM
Total Risk = E(V(X given theta)) + V(E(X given theta))

Question 20

Name the 3 models Meyers tested on Incurred Data

Answer 20

1. Mack (Light-Tail)
2. LCL (Light-Tail)
3. CCL (Validated, mildly light tail)

Question 21

Name the 6 models Meyers tested on Paid Data

Answer 21

1. ODP (Biased High)
2. Mack (Biased High)
3. CCL (Biased High)
4. LIT (Biased High)
5. CIT (Biased High)
6. CSR (Validated)

Question 22

State 2 reasons that might explain model produces expected loss estimates that are biased high when modelling paid losses.

Answer 22

1. Ins Loss environment has experienced changes that are not observable at current time.
2. There are other models that can be validated.

Question 23

Describe 2 ways to increase the variability of the predictive distribution

Answer 23

1. Treat level of each AY as random (ex: LCL)
2. Allow for more correlation between Ays (ex: CCL)

Question 24

Which of LCL and CCL produce the highest standard deviation.

Answer 24

CCL since it includes correlation parameter which increases variability.

Question 25

Is LCL and CCL standard deviation higher or lower than Mack?

Answer 25

Higher since they predict more risk.

Question 26

2 consequences of inclusion of payment year trend in model

Answer 26

1. Model should be based on incremental paid loss amounts rather than cumulative paid loss amounts. This is because cum loss includes settle claims which do not change with time. 2. Incremental paid loss amounts tend to be skewed to the right and are occasionally negative. We need a loss distribution that allows for these features (ex: skew normal)

Question 27

Compare relationship between sigma_d and d in CIT vs CCL

Answer 27

Since CCL is applied to cum losses, s_d decreases as d increases because greater proportion of claims are settled (less variability). Since CIT is applied to incremental losses, s_d increases as d increases because smaller, less volatile claims tend to be settled earlier.

Question 28

Contrast LIT and CIT

Answer 28

LIT is similar to CIT, but it does not include AY correlation. LIT produce worser results than CIT. Neither show improvement over ODP or Mack.

Question 29

True or False? Meyers conclusions apply to any dataset.

Answer 29

False Conclusions apply specifically to the dataset studied by Meyers. Other datasets may exhibit different behaviour.

Question 30

Describe 2 advantages of Correlated Chain Ladder method

Answer 30

1. Incorporates correlation between accident years 2. Treats the level of each accident year as random, which allows it to predict more risk

Question 31

Describe 2 advantages of the Bayesian method

Answer 31

1. Incorporates expert opinion into the simulation 2. Produces full predictive distribution rather than just a point estimate