Meyers

Question 1

When models do not accurately predict dist of outcomes for test data, 3 explanations

Answer 1

1. Insurance process is too dynamic to be captured by single model
2. Could be other models that better fit data
3. Data used to calibrate model is missing crucial info needed to a make reliable prediction

Question 2

3 tests to validate models

Answer 2

1. histogram
2. p-p plot
3. K-S statistic

Question 3

histogram

Answer 3

• if percentiles are uniformly distributed, height of bars should be equal
• for small sample, not perfectly level
• if level, model is appropriate

Question 4

p-p plot

Answer 4

• tests for stat significance of uniformity
• plot expected percentiles on x and sorted predicted percentiles on y -> if predicted percentiles are uniformly dist, plot lies along 45 degree line

ie model is appropriate if p-p plot lies along 45 degree line

expected value e = {1/(n+1),…,n/(n+1)}

Question 5

K-S statistic

Answer 5

D=max|pi-fi|

fi = 100*{1/n,…,n/n}

• can reject hypothesis that set of percentiles is uniform @ 5% level if D > critical value = 136/sqrt(n)
• critical values appear as 45 degree bands that run parallel to y=x
• Meyers deems model validated if passes K-S test

Question 6

Validating Mack: results

Answer 6

• incurred data
• on histogram, percentiles show little uniformity and actual outcomes are falling into smaller and larger percentiles more often -> Mack produces dist that is light tailed
• in p-p plot, predicted percentiles form S shape -> light tailed because actual outcomes failing into percentiles that are lower than expected in left tail and higher in right tail
• D > critical value

Question 7

Validating ODPB: results

Answer 7

• paid data
• predicted outcomes are occurring in lower percentiles more often -> implies both models produce expected loss estimates that are biased high when modeling paid losses
• producing higher expected loss estimates, left tail becomes lighter
• D > critical value

Question 8

Possible reasons for observations for paid and incd data ie Mack and ODPB results

Answer 8

• insurance loss environment has experience changes that are not yet observable
• other models that can be validated

Question 9

Bayesian models for Incurred loss data

Answer 9

-Mack model underestimates variability of predictive distribution which leads to light tails

Leveled Chain Ladder (LCL)

Correlated Chain Ladder (CCL)

Question 10

Leveled Chain Ladder (LCL)

Answer 10

• treats level of AY as random ie independence between AY -> model will predict more risk
• sigma is larger for earlier DPs where more claims open and more variability

Question 11

Correlated Chain Ladder (CCL)

Answer 11

• allows for correlation between AYs -> model will predict more risk than LCL
• should result in larger standard deviation for predicted distribution (heavier in tails), which would result in percentiles of outcomes to be more uniform than LCL

Question 12

LCL results

Answer 12

• produce higher std dev than Mack
• has S shape and some points lie outside K-S bounds, but improvement over Mack, & D is closer to critical value

Question 13

CCL results

Answer 13

• produce higher std dev than Mack
• CCL produced higher std dev for each AY than LCL
• CCL has S shape and all points within bounds and D is smaller than critical value -> model validates against data and exhibits uniformity

Question 14

Bayesian models for Paid loss data

Answer 14

-CCL model produced estimates that were biased high

Correlated Incremental Trend (CIT)

Leveled Incremental Trend (LIT)

Question 15

Correlated Incremental Trend (CIT)

Answer 15

• introduces payment trend and dist is skewed right and allows for negative values (model should be based on increm paid)
• sigma is smaller for earlier DPs
• opposite from LCL b/c increm loss

Question 16

Leveled Incremental Trend (LIT)

Answer 16

-similar to CIT but does not have AY correlation

Question 17

results for CIT and LIT

Answer 17

• CIT and LIT produce estimates that are biased high
• neither show noticeable improvement over ODP and Mack

Question 18

Changing Settlement Rate (CSR)

Answer 18

• claims are reported and settled faster due to tech and CIT model might not fully reflect this change
• allows for changing settlement rates which can reflect speedup in claim settlement for more recent AY
• cum. paid losses since no longer considering payment yr trend

Question 19

CSR results

Answer 19

• histogram is nearly level
• p-p plot closely tracks with y=x
• indicates that incurred data recognized speedup in claims settlement rate which led to good fit with CCL

Question 20

total risk

Answer 20

total risk = process risk + parameter risk

Question 21

process risk vs parameter risk

Answer 21

• process risk = avg var of outcomes from expected result
• parameter risk = var due to many possible parameters in posterior dist of parameter

Question 22

Meyers found what risk is close to total risk for several insurers

Answer 22

parameter risk

Question 23

model risk

Answer 23

• model risk = risk that we did not select right model
• model risk is special case of parameter risk

Question 24

if p-p plot shows S curve

Answer 24

demonstrates more high and low percentiles than expected

45 degree = uniformly distributed

Question 25

CCL: distribution for ultimate loss

Answer 25

start with loss given C(w,d) and then calc u(w,d) using below

u(1,d)=alpha(1)+beta(d)

calculate parameters of distribution for ~C by correlating with AY given:

~u(w,d)=alpha(w)+beta(d)+row*(ln(C(w-1,d))-u(w-1,d))

C(w-1,d) and u(w-1,d) is from 1 steps

~C(w,d) is simulated from lognormal with log mean ~u(w,d) and log std dev σ(d)

Question 26

CSR: distribution for ultimate losses

Answer 26

u(w,d)=alpha(w)+beta(d)*(1-gamma)^(w-1)

C(w,d) is simulated from lognormal dist with log mean=u(w,d) and log std dev=σ(d)

Question 27

CSR: how gamma parameter impacts claims payment pattern

Answer 27

development period portion of logmean formula is Beta(d)*(1-gamma)^(w-1)

with later AY, absolute value will be smaller

the larger this portion is, the larger logmean, resulting in higher simulated losses

If Beta(d) is negative, then logmean larger

higher simulated losses indicate speedup in settlement rate for more recent AY

Question 28

LCL compared to MACK

Answer 28

-compared to Mack, model uses random level parameters instead of fixed level parameters for most recent cumulative loss for the AY

Question 29

CCL compared to MACK

Answer 29

• model uses random level parameters instead of fixed level parameters for most recent cumulative loss for the AY
• model incorporates correlation between AY

Question 30

procedure used by CCL to create loss distribution for ultimate losses

Answer 30

1. use loss triangle and prior distributions for CCL model, run MCMC script to estimate posterior distributions
2. create sample sets of parameters from posterior distributions
3. for each parameter set, simulate the ultimate losses, iterated for each AY
   - simulated losses C(w-1,10) is used to calc u(w) for next AY; simulated C(w,10) is then simulate with u(w) and sigma(10)
4. distribution and any summary statistics are calculated off of total ultimate losses across AY

Question 31

incorporating expert knowledge about expected losses in CCL

Answer 31

prior distribution for level parameters and logelr parameter can be specified to be more restrictive instead of using vague priors so that model better reflects expected losses