Section A: Meyers, Taylor & Marshall

Question 1

Taylor: What are the results of Taylor’s Theorems?

Answer 1

Theorems 3.2 and 3.3 are similar to 3.1 about the ODP Mack model and mean:

1. Forecasts from the ODP Mack and ODP Cross-Classified models are identical and the same as those from the chain ladder method despite the different formulations.
2. We can get forecasts for the ODP Cross-Classified model without considering the model directly and working as if the model was an ODP Mack model.

Question 2

Marshall: Preparing the Claims Portfolio:

Considerations for Splitting into Valuation Classes

Answer 2

• Preferable to split the claims portfolio into valuation classes the same as the split used to derive the central estimates
  
  • This allows analyzed sources of uncertainty to be aligned with the central estimate analysis.
• If the valuation of the central estimate is at a granular level, it may make sense to do the quantitative analysis on aggregated valuation classes (which are more credible) and allocate results down.

Question 3

Taylor: Tweedie Sub-Family

Expected Value & Variance

Answer 3

Question 4

Taylor: ODP Cross-Classified Model GLM Representation (for a 4 x 4 triangle)

Answer 4

Question 5

Taylor: The expected value and variance of an EDF:

E[Y] = ?

Var[Y] = ?

Answer 5

E[Y] = µ = b’(𝜃)

Var[Y] = a(ɸ)b’‘(𝜃)

Question 6

Taylor: The Exponential Dispersion Family (EDF) is the family of distributions with probability density function (pdf) π(y;𝜃,ɸ)of the form:

ln(π(y;𝜃,ɸ)) = ?

Answer 6

ln(π(y;𝜃,ɸ)) = [y𝜃 - b(𝜃)] / a(ɸ) + c(y,ɸ)

y = value of observation Y

𝜃 = location parameter called canonical parameter

ɸ = dispersion parameter called scale parameter

b(•) = cumulant function which determines shape of distribution

e^c(y,ɸ) = normalizing factor producing unit total mass for the distribution

Question 7

Meyers: Bayesian Loss Models

Correlated Incremental Trend (CIT) Formulas

Answer 7

µ_w,d= ⍺_w+ β_d + 𝜏(w + d - 1)

Z_w,d ~ lognormal(µ_w,d, σ_d)

IncLoss_1,d^sim ~ normal(Z_1,d, 𝝳)

IncLoss_w,d^sim ~ normal(X_w,d + ⍴(IncLoss_w-1,d^sim - Z_w-1,d)e^𝜏,𝝳) for w>1

• LIT model is the same, but sets ⍴ = 0

Question 8

Marshall: Calibrating balance scorecard scores to CoVs

Example Scale and Comments

Answer 8

Question 9

Meyers: Non-Bayesian Models

Briefly describe how the ODP Bootstrap model on paid data performed in the Meyers paper?

Answer 9

• Histogram shows more low percentiles than expected
• p-p plot shows a slanted “U” curve
• K-S test fails at the 5% level for all the lines combined

Conclusion

The ODP Bootstrap model is biased high

Question 10

Meyers: Describe three tests for uniformity for n predicted percentiles.

Answer 10

1. Histogram - if percentiles are uniformly distributed, the bars should be of similar height
2. p-p Plot - sort the predicted percentiles into increasing order, then plot the expected percentiles on x-axis and sorted predicted on y axis.
   
   • sorted predicted percentiles should follow the expected percentiles along the 45 degree line
   • {e_i} = 100*{1/(n+1), 2/(n+1),….,n/(n+1)}
3. K-S Test - reject hypothesis of uniformity if D>D* at 5% level of significance
   
   • D = Max | p_i - f_i |
   • Critical Value = 136 / √n
   • f_i = 100 * {1/n, 2/n,….,n/n} where n is the # of percentiles
   • On a p-p plot, these critical value appear as 45 degree bands that run parallel to the line y=x. We reject the hypothesis of uniformity if the p-p plot lies outside the bands.

Question 11

Marshall: What is the goal of External Benchmarking?

Answer 11

Goal - compare the reasonableness of CoVs and risk margins to those from external sources.

Should NOT be relied on instead of analysis, but may be useful when there is little information available for analysis; especially for independent risk.

Question 12

Marshall: External Systemic Risk

Latent Claim Risk

Answer 12

Latent Claim Risk

• Negligible for most valuation line classes but material for some Workers Compensation and liability classes
• Low probability risk with very high potential severity (e.g. Asbestos)
• If the risk exposure is significant enough to be included in the central estimate valuation or capital adequacy modeling, then it should be examined more closely for setting risk margins
• Discuss with underwriters to identify sources of latent claim risk

Question 13

Marshall: Preparing the Claims Portfolio:

Allocation of Valuation Classes to Claim Groups

Answer 13

• If different groups of claims within a valuation class have materially different uncertainty, they should be treated separately in the risk margin analysis
  
  • e.g Splitting the Home portfolio between CAT and non-CAT claims, splitting non-CAT between liability and non-liability
• Balance the benefit gained and the practicality/cost

Question 14

Marshall: External Systemic Risk

Claim Management Process Change Risk

Answer 14

Claim Management Process Change Risk

• Typically impacts all valuation classes with different levels of materiality
• Discuss current and future process changes with claims management
  
  • Consider impact on reporting/payment patterns, closing and reopening rates, and case estimation
• Could do sensitivity testing of key valuation assumptions to help assess CoV for this category
• Likely to be more relevant for OSC than premium liabilities

Question 15

Taylor: The standardized Pearson residuals are defined as follows:

R_i^P = ?

Answer 15

R_i^P = (Y_i - Yhat_i) / σhat_{<strong>i</strong>}

Check residual plots against each variable:

• Standardized Pearson residuals should be random about zero (unbiased) and have uniform dispersion (homoscedasticity)

Question 16

Marshall: Independent Risk Assessment for Oustanding Claim Liabilities

Answer 16

Goal - a model is used to “fit away” past systemic episodes in order to analyze the residual volatility and get the CoVs for independent risk.

Methods Used

• Mack Method
• Bootstrapping
• Stochastic Chain Ladder
• GLM Techniques
• Bayesian Techniques

Question 17

Marshall: External Systemic Risk

Event Risk

Answer 17

Event Risk

• Catastrophes have most material impact on Property
• Include in OSC liability if there are outstanding events
• Information to help quantify risk for premium liabilities:
  
  • past experience from event risk - could create frequency/severity model and then adjust past experience for changes in portfolio/inflation/…
  • CAT models which produce a range of possible outcomes for modelled events on an insurer’s book
  • Reinsurance intermediaries

Question 18

Marshall: External Sytemic Risk CoV Consolidation Formula

Answer 18

CoV²_{<em>external</em>} = ∑CoV_i²

• Formula should be similar to internal as some external risks are correlation (within the same class or between PL and CL)

Question 19

Taylor: ODP Mack Model GLM Representation

Answer 19

Question 20

Marshall: Summary of Risk Margin Analysis Framework

Answer 20

Question 21

Marshall: Describe the internal benchmarking that should occur for the following source of risk:

Independent Risk

Answer 21

Portfolio Size: larger portfolios have lower CoVs due to lower volatility from random effects

Length of Claim Runoff: Longer tailed lines have higher CoVs due to more time for random effects to have an impact

Implications:

Long-tailed portfolios - premium liability CoV should be higher than claim liability CoV

Short-tailed portfolios - premium liability CoV should be lower than claim liability CoV

Question 22

Marshall: What are the sources of Systemic Risk?

Answer 22

Internal Systemic Risk

Risks internal to the liability valuation process

→ Reflects the extent to which the actuarial valuation approach is an imperfect representation of complex, real-life process

External Systemic Risk

Risks external to the actuarial modeling process

→ Even if the model represent the current conditions well, future systemic trends may cause future experience to differ from current expectations

Question 23

Taylor: What do you need to do to specify a GLM?

Answer 23

The selection of a GLM consists of the following:

• Selection of a cumulant function, controlling the model’s error distribution
• Selection of an index p, controlling the relationship between the models means and variance
• Selection of the covariates x_i^T
  
  • the variables that explain u_i
• Selection of link function, controlling the relationship between the mean ui and the associated covariates

Question 24

Meyers: Interpreting Diagnostics

Light-Tail Model

Answer 24

Question 25

Marshall: External Systemic Risk ## Footnote **Recovery Risk**

Answer 25

**Recovery Risk** * Likely to be insignificant except possible for Auto 3rd party recoveries * Non-Reinsurance recovery rates can be analyzed to understand past systemic risks. Discuss potential risks with Claims management. * Reinsurance Recoverability * Systemic risk of recovery could be driven by CATs or falling asset returns * Discuss with reinsurance management to identify possible scenarios, likelihood and potential impact

Question 26

Marshall: Mechanical Hindsight Analysis Steps

Answer 26

**_Step 1_** - apply the chain ladder method to the data as of the valuation date to calculate the current unpaid estimate. **_Step 2_** - iteratively remove a diagonal of data and apply the same method (and average age-to-age factors) to calculate unpaid estimates for prior valuation dates. **_Step 3_** - Compare the past estimates to the current estimate for the same accident years (similar to a runoff). The relevant payments made between the past valuation dates and the current valuation date should be added to the current unpaid estimate.

Question 27

Taylor: What is a consquence of using Pearson residuals? How is this overcome?

Answer 27

One consequence of Pearson residuals is that they will reproduce any **non-normality** that exits in the observations (eg. skewed loss data). Normally distributed residuals are more useful for model assessment, so its common to look at **standardized deviance** residuals when assessing a GLM

Question 28

Taylor: Results of the Mack Model

Answer 28

Results of the Mack Model: 1. The conventional chain ladder LDF estimators, f_j^hat are: * Unbiased * Minimum variance among estimators that are unbiased linear combinations of the age-to-age factors, f_j^hat 2. The conventional chain ladder reserve estimator, R_k^hat, is unbiased.

Question 29

Taylor: Theorem 3.3 Regarding Cross-Classified Model

Answer 29

If the Theorem 3.2 assumptions hold and the fitted values of Y_k,j^hat and reserve estimates R_k^hat are correct for bias, then they are MVUEs of Y_k,j and R_k.

Question 30

Marshall: External Systemic Risk ## Footnote **Legislative, Political and Claim Inflation Risk**

Answer 30

**Legislative, Political and Claim Inflation Risk** → more material for long-tail classes and sub-risks are often correlated _Sub-groups include:_ * Impact of recent legislation and court interpretations * Potential future legislation with retrospective impacts * Precedent setting in courts * Changes in medical technology costs * Change in legal costs * Systemic shifts in large claim frequency/severity

Question 31

Marshall: What is the _goal_ of Internal Benchmarking?

Answer 31

Goal: check for internal consistency and reasonableness by comparing CoV * between similar valuation lines * between claim and premium liabilities within classes

Question 32

Marshall: Describe the internal benchmarking that should occur for the following source of risk: ## Footnote **Internal Systemic Risk**

Answer 32

**_Compare CoVs by valuation class:_** * If template models are used for similar valuation classes, we would expected similar CoVs. * i.e. classes with homogenous claim groups should have similar CoVs * If similar valuation methodology is used on both short and long-tail classes, then we would expect a higher CoV for the long-tail class.

Question 33

Meyers: Bayesian Paid Loss Models Describe the _Correlated Incremental Trend (CIT)_ model and how it performed on **paid loss** data analyzed in the paper.

Answer 33

**_Key Improvement_** * CIT includes a calendar year trend for incremental paid loss * CIT allows for correlation between accident years (like CCL) * Histogram and *p-p* plot show more low percentiles than expected * K-S test _fails_ at the 5% level for all lines combined **_Conclusion_** The CIT model is _biased high on paid losses._

Question 34

Marshall: Independent Risk CoV consolidation Formula ## Footnote **CoV_ind = ?**

Answer 34

σ²_ind = ∑σ_i²*w*_i² CoV_ind = [(∑CoV_i • *weight_i*)²]^1/2 * assume no correlation between Home/Auto (classes) * weights sum up to 1 when looking at totals CoV_i = [∑(CoV_i•weight_i)²]^1/2 / ∑w_i

Question 35

Marshall: Internal Systemic Risk ## Footnote **Balance Scorecard Approach**

Answer 35

**_Goal_** - assess the adequacy of the modeling infrastructure and its ability to reflect and predict the underlying insurance process. **_Steps:_** 1. **Conduct a qualitative assessment** of the modeling infrastructure for each specification/parameter/data risk component (score between 1 and 5) 2. **Calculate a weighted average score** using weights for each risk indicator that reflect the risks relative importance. 3. **Calibrate** weighted average scores to a CoV scale for internal systemic risk.

Question 36

Meyers: Bayesian Incurred Loss Models Describe the _Level Chain Ladder (LCL)_ model and how it performed on **incurred loss** data analyzed in the paper.

Answer 36

**_Key Improvement over Mack_** * LCL uses **random level parameters** for each AY µ_w,d = ⍺_w + β_d Loss_w,d^sim ~ lognormal (µ_w,d, σ_d) * Histogram and *p-p* plot show more high/low percentiles than expected * K-S test fails at the 5% level for all lines combined **_Conclusion_** LCL is still too light in the tails, but better than Mack

Question 37

Marshall: Briefly describe general hindsight analysis.

Answer 37

**Goal** - compare the past reserve estimates of liabilities against the latest view of the same liabilities to _review the actual volatility in the past_. **_Considerations_** * Valuation models may have changed from previous valuations * May be different sources of external systemic risk in the future * Less valuable for long-tail portfolio with serial correlations between consecutive valuations

Question 38

Taylor: What does Taylor suggest for correcting for heteroscedasticity?

Answer 38

* if a residual plot exhibits heteroscedasticity, then **weights** can be used to correct it. * **_General Rule_** - assign weights that are inversely proportional to the variance of the residuals * e.g. residuals above age 55 have double the standard deviation of those below age 55 so this means the residuals above age 55 have 4 times the variance than those below (weight would be 1/4)

Question 39

Marshall: Describe Scenario Testing

Answer 39

Goal: identify scenarios (e.g. higher frequency or severity, loss ratios, etc.) that would result in the central estimate **increasing to the same level as the risk-loaded actuarial central estimate.** * how key assumptions underlying the central estimate calculation would need to change in order to produce a central estimate equal to the risk-loaded actuarial central estimate

Question 40

Marshall: Internal Systemic Risk CoV Consolidation Formula

Answer 40

CoV_internal = **[**(CoV_i•*w*_i)² + (CoV_j•*w_j*)² + 2⍴_ij*w*_i*w*_jCoV_iCoV_j**]**^1/2 * same as the variance formula with correlation

Question 41

Meyers: Briefly describe the reasons why models may not accurately predict distributions of actual outcome?

Answer 41

* Insurance losses are **too dynamic** to be fully reflected in a single stochastic model * other models may prove to be a better fit * Data used in the model might be **missing key information** * difficult to make accurate estimates if data is missing (e.g. changes in underlying business, claims process or reinsurance)

Question 42

Marshall: External Systemic Risk ## Footnote **Economic and Social Risk**

Answer 42

**_Economic & Social Risk_** * Standard inflation * Unemployment * GDP growth * Interest rates * Asset returns * Fuel prices * Changing driving patterns * Systemic shifts in claim frequency for short-tail lines

Question 43

Marshall: Describe the internal benchmarking that should occur for the following source of risk: ## Footnote **External Systemic Risk**

Answer 43

CoVs should be _higher for long-tail portfolios_, except for event risk for Property and liability risk for Home.

Question 44

Meyers: Total Risk =

Answer 44

Var(X) = E_𝜃[Var[X|𝜃]] + Var_𝜃[E[X|𝜃]] Total Risk = Process Risk + Parameter Risk

Question 45

Meyers: Non-Bayesian Models Briefly describe how the Mack model performed on the _paid loss data_ in the Meyers paper?

Answer 45

* Histogram shows more low percentiles than expected * p-p plot shows a slanted "U" curve * K-S test fails at the 5% level for all the lines combined **_Conclusion_** Paid Mack model is _biased high_

Question 46

Marhsall: Total Consolidated CoV Formula CoV*_total*

Answer 46

CoV_total = **(**CoV_ind² + CoV_internal² + CoV_external²**)**^1/2

Question 47

Marshall: Describe Sensitivity Testing

Answer 47

**Goal:** gain insights about the sensitivity of the final risk margin to key assumptions **_Steps:_** 1. Independently, vary key assumptions (the CoVs and correlations) and then monitor the impact on the risk margin 2. Review key assumptions that have significant impact on the risk margin

Question 48

Marshall: Why is both quantitative and qualitative analysis necessary to properly assess risk margins?

Answer 48

* Quantitative analysis can only reflect uncertainty in historical experience and can't capture adequately all possible sources of future uncertainty * Judgement is necessary to estimate future uncertainty

Question 49

Taylor: Re-Normalized ODP Cross-Classified Model Parameters

Answer 49

Question 50

Marshall: What sources of uncertainty can quantitative modeling best assess?

Answer 50

* Quantitative modeling is best for analyzing **independent risk and past episodes of external systemic risk.** * Quantitative modeling must be _supplemented_ with other qualitative or quantitative analysis to incorporate internal systemic risk and external systemic risk. * Future external systemic risk may differ from past episodes

Question 51

Marshall: What are the main sources of Internal Systemic Risk?

Answer 51

**_Specification Error_** Error arising from an inability to build a model that fully represents the underlying insurance process. **_Parameter Selection Error_** Error that the model can't adequately measure all predictors of claim cost outcomes or trends in predictors. May be more cost drivers than can be captured in the valuation model. **_Data Error_** Error from poor data or unavailability of data required for a credible valuation model.

Question 52

Marshall: Risk Margin Formula Lognormal Distribution

Answer 52

Risk Margin (%) = **[**e^[µ+Zσ] / (*w*₁ + *w*₂)**]** - 1

Question 53

Marshall: What are the sources of uncertainty?

Answer 53

**_Systemic Risk_** Risks that are potentially _common_ across valuation classes or claim groups **_Independent Risk_** Risks arising from _randomness_ inherent in the insurance process

Question 54

Taylor: Theorem 3.1 Regarding the EDF Mack Model

Answer 54

Theorem 3.1: a. If the original Mack variance assumption also holds, then the MLEs of the f_j parameters are the chain ladder LDF estimators, f_j^hat, and these are unbiased estimators. b. If the model is restricted to the ODP Mack model and if the dispersion parmeters are just column dependent, ɸ_k,j = ɸ_j, then the weighted average chain ladder LDFs, f_j^hat, are MVUEs. c. For the same conditions as (b), the cumulative loss estimates, X_k,j^hat, and reserve estimates, R_k^hat, are also MVUEs.

Question 55

Meyers: Interpreting Diagnostics Validated Model

Answer 55

Question 56

Taylor: Describe Continuous Covariates.

Answer 56

Continuous covariates are predictors with continuous levels such as age * represented by polynomial * transformed these variables before they go into the model (e.g. apply a log transformation)

Question 57

Meyers: Interpreting Diagnostics Biased High Model

Answer 57

Question 58

Meyers: What is parameter risk?

Answer 58

**_Parameter Risk_** Variance due to the uncertainty in the parameters, reflected in the posterior distributions of the parameters * Parameter risk represents the overwhelming majority of the total risk in the loss data sets that Meyers looked at.

Question 59

Marshall: What are the correlation effects on internal systemic risks?

Answer 59

Assumed to be **uncorrelated** with independent risk and external systemic risk sources. There is **correlation** between classes and between outstanding claim and premium liabilities due to: * _Same actuary effect_ - effect of common valuation models or approaches across different valuation lines * _Linkage_ between *premium liability* methodology and outcomes from *outstanding claims* valuation

Question 60

Marshall: Why quantitative method might **not** be appropriate for assessing correlation effects

Answer 60

* Techniques tend to be complex and require substantial data * Time/effort required may outweigh the benefits * Correlations would be heavily influenced by past correlations * Difficult to _separate_ past correlation effects between independent risk and systemic risk or identify the effects of past systemic risks * Internal systemic risk can't be modelled with standard correlation risk techniques * Results unlikely to be aligned with framework, which splits between independent, internal systemic, and external systemic risk

Question 61

Taylor: What is the implication of Theroem 3.1?

Answer 61

This theorem means that the conventional chain ladder estimates and forecasts are optimal estimators (both MLE and MVUEs). This is stronger implication than the original Mack model →It shows the LDF estimators, f_j^hat, are minimum variance for all unbiased estimators, not just of the linear combinations f the age-to-age factors.

Question 62

Meyers: Non-Bayesian Models Briefly describe how the Mack model performed on the _incurred loss data_ in the Meyers paper?

Answer 62

* Histogram shows more high/low percentiles than expected * p-p plot shows a slanted "S" curve and the K-S test failed at the 5% level for all the lines combined **_Conclusion_** Incurred Mack model is _too light_ in the tails

Question 63

Taylor: Non-Parametric Mack Model Assumptions

Answer 63

* Accident years are stochastically independent * For each AYk, the cumulative losses Xkj form a Markov chain * For each accident year k and development peirod j: * E[X_k,j+1|X_k,j] = f_jX_k,j * Var[X_k,j+1|X_k,j] = σ²_jX_k,j

Question 64

Meyers: What is process risk?

Answer 64

**_Process Risk:_** Average variance of outcomes from the expected result

Question 65

Marshall: External Systemic Risk: ## Footnote **List the Risk Categories**

Answer 65

* Economic & Social Risks * Legislative, Political Risks and Claim Inflation Risks * Claim Management Process Change Risk * Expense Risk * Event Risk * Latent Claim Risk * Recovery Risk

Question 66

Marshall: What are the correlation effects on external systemic risks?

Answer 66

Uncertainty from each risk category is assumed to be _uncorrelated_ with independent risk, internal systemic risk and the uncertainty from other external systemic risk categories. There is _correlation_ between classes (home & auto) and between outstanding claim and premium liabilities from similar risk categories * e.g. claims inflation risk across all long-tail portfolios

Question 67

Marshall: Internal Systemic Risk ## Footnote **Potential Risk Indicators**

Answer 67

**_Specification Error_** * Number of independent models used * Range of results produced by models * Checks made on reasonableness of results **_Parameter Selection Error_** * Best predictors have been identified * Best predictors are stable over time (or change due to process changes) * Value of predictors used (close to best predictors) **_Data Error_** * Good knowledge of past process affecting predictors * Extent, timeliness, consistency and reliability of information from business * There are appropriate reconciliations and quality control for the data

Question 68

Marshall: External Systemic Risk ## Footnote **Expense Risk**

Answer 68

**Expense Risk** * Generally a small contributor to external systemic risk * Discuss with product/claims management to understand the drivers of policy maintenance and claim handling expenses * Can do sensitivity testing around key drivers * Relevant to Premium Liabilities (paper mentions only PL (unexpired portion of policies) not CL)

Question 69

Taylor: GLM Deviance

Answer 69

**D\*(Y, Yhat) = 2(loglikihood_saturated - logliklihood_Model)** ## Footnote The saturated model has a parameter for each observation so the model completely fits the observations.

Question 70

Marshall: What are the sources of Independent Risk?

Answer 70

* Random component of _parameter risk_ * Randomness of the insurance process compromises ability to select appropriate parameters for valuation models * Random component of _process risk_ * Pure effect of randomness of the insurance process

Question 71

Taylor: How do you determine all the expected future increamental losses at once?

Answer 71

* Use a forecast design matrix (X) Yhat\* = µhat\* = h^-1(X\*βhat)

Question 72

Marshall: Calibrating balance scorecard scores to CoVs ## Footnote **Developing a Scale**

Answer 72

* Scale should _represent uncertainty_ of internal systemic risk ranging from worst to best practice * Relative difference in performance of modeling infrastructures over time can give insight into uncertainty of poor vs. good modeling * Minimum CoV for best practice is unlikely to be \<5% * the best models can't completely model the insurance process * Might have CoV\>20% for single, aggregated model with limited data

Question 73

Marshall: Risk Margin Formula ## Footnote **Normal Distribution**

Answer 73

Risk Margin (%) = z•CoV_total

Question 74

Marshall: Framework for Assessing Risk Margin

Answer 74

Question 75

Meyers: Interpreting Diagnostics Heavy-Tail Model

Answer 75

Question 76

Taylor: Describe Categorical Covariates.

Answer 76

Categorical Covariates are predictors with discrete levels * e.g. state or province, gender, make or model of car * each level is treated as dummy variable (values of 0 or 1) * e.g. Alberta, Manitoba and Saskatchewan are 3 provinces in the dataset the model is fit to. If the policyholder lives in Alberta then the covariate would be 1 0 0

Question 77

Marhsall: What are the correlation effects on independent risks?

Answer 77

Assumed to be **_uncorrelated_** with any other source of uncertainty either within or between valuation classes

Question 78

Taylor: How can the scale parameter be estimated using deviance?

Answer 78

ɸhat = D\*(Y,Yhat) / (n-p) ## Footnote n = number of observations p = number of parameters

Question 79

Meyers: What is model risk?

Answer 79

**_Model Risk_** The risk that we didn't selected the "correct" model. * Can think of model risk as a special case of parameter risk when we weight together different models. The weights used would be considered parameters. * If the posterior distribution of the weights assigned to each model has significant variability then model risk exists. * Quantification will show up in process risk portion of the total risk.

Question 80

Meyers: Summary of Model Conclusions

Answer 80

Question 81

Taylor: Parametric Mack Model Assumptions

Answer 81

Same assumptions as Mack Model except: * Y_k,j+1 | X_k,j ~ EDF(𝜃_k,j, ɸ_k,j;a,b,c) * Variance assumption is removed - Variance is driven by the selected EDF distribution

Question 82

Taylor: Theorem 3.2 Regarding the Cross-Classified Model

Answer 82

For the cross-classified model, if the following assumptions hold: * Y_k,j is restricted to an ODP distribution * The dispersion parameters are identical for all cells: ɸ_kj = ɸ Then the MLE fitted values Y_k,j^hat are the **same** as those from the conventional chain ladder method.

Question 83

Taylor: Cross-Classified Model Assumptions

Answer 83

* The random variables Y_k,j are stochastically independent * For each accident year k and development period j * Y_k,j+1 | X_k,j ~ EDF(𝜃_k,j, ɸ_k,j; a,b,c) * E[Y_k,j] = ⍺_kβ_j * Σβ_j = 1

Question 84

Meyers: Bayesian Incurred Loss Models Describe the _Correlated Chain Ladder (CCL)_ model and how it performed on **incurred loss** data analyzed in the paper.

Answer 84

**_Key Improvement over LCL_** * CCL allows for **correlation** between accident years µ_1,d= ⍺₁+ β_d µ_w,d= ⍺_w+ β_d+⍴(ln(C_w-1,d) - µ_w-1,d) for w\>1 Loss_w,d^sim ~lognormal(µ_w,d, σ_d) * ⍴ is generally positive which results in higher prediction variance than LCL * Histogram and *p-p* plot show slightly more high/low percentiles than expected and the K-S test _passes_ at the 5% level. **_Conclusion_** The CCL model is validated, but has mildly thin tails. *Note - since correlation & random level parameters is added to the model the variability increases and is a better fit compared to Mack and LCL.*

Question 85

Taylor: How are parameters of a GLM usually estimated?

Answer 85

Parameters of a GLM are often estimated using maximum likelihood estimation (MLE)

Question 86

Meyers: Bayesian Paid Loss Models Describe the _Changing Settlement Rates (CSR)_ model and how it performed on **paid loss** data analyzed in the paper.

Answer 86

**_Key Improvement_** * CSR parameter ɣ reflects changes to the **claim settlement rate** µ_w,d= ⍺*_w*+ β_d • (1-ɣ)^w-1 Loss*_w,d^sim* ~ lognormal(µ_w,d,σ*_d*) * ɣ is generally _positive_, reflecting a speedup in payment pattern * Histogram and *p-p* plot show CSR _corrects_ the high bias or other Bayesian paid models and the K-S test _passes_ at the 5% level of confidence **_Conclusion_** The CSR model is validated.

Question 87

Taylor: Standardized Deviance Residuals R_i^D = ?

Answer 87

**R_i^D = sgn(Y_i - Yhat_i)(d_i/ɸhat)^0.5** ## Footnote d_i = the contribution of the ith ovservation to the deviance D\*(Y,Yhat) sgn = -1 if quantity is negative, 0 if quantity is zero, and 1 if quantity is +1

Question 88

Taylor: ODP Cross-Classified Model Parameters Formulas

Answer 88