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What is the formula used to calculate α3 in Step 7?
What is the formula used to calculate α3 in Step 7?
In Step 8, how is β1 calculated?
In Step 8, how is β1 calculated?
What does the equation 'αk' represent in this context?
What does the equation 'αk' represent in this context?
What is indicated if the sum of the βj parameters does not equal 1?
What is indicated if the sum of the βj parameters does not equal 1?
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What is the final value of β1 as calculated in Step 8?
What is the final value of β1 as calculated in Step 8?
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Which step involves checking the sum of the βj parameters?
Which step involves checking the sum of the βj parameters?
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What is the purpose of dividing each βj by their sum if an adjustment is necessary?
What is the purpose of dividing each βj by their sum if an adjustment is necessary?
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What is implied by the incremental estimates being equal using different models?
What is implied by the incremental estimates being equal using different models?
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What does q(k, j) represent in the calculation process?
What does q(k, j) represent in the calculation process?
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In the cross-classified model, how is q(3, 3) calculated?
In the cross-classified model, how is q(3, 3) calculated?
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What does the formula fj calculate in the additional calculation?
What does the formula fj calculate in the additional calculation?
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What is the advantage of using Generalized Linear Models (GLMs) in this context?
What is the advantage of using Generalized Linear Models (GLMs) in this context?
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In the design matrix X for a GLM, what do the rows represent?
In the design matrix X for a GLM, what do the rows represent?
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How is the cumulative loss estimate adjusted in the chain ladder method?
How is the cumulative loss estimate adjusted in the chain ladder method?
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Which of the following calculations represents the correct value of fj when summing the β values?
Which of the following calculations represents the correct value of fj when summing the β values?
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What does the term maximum likelihood indicate in the context of chain ladder results?
What does the term maximum likelihood indicate in the context of chain ladder results?
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Which distribution is commonly used for modeling loss amounts?
Which distribution is commonly used for modeling loss amounts?
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What does the notation $b'(θ)$ represent in the context of the distributions?
What does the notation $b'(θ)$ represent in the context of the distributions?
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In the Poisson distribution, what is the variance $Var(Y)$ equal to?
In the Poisson distribution, what is the variance $Var(Y)$ equal to?
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Which of the following statements about the variance function is accurate?
Which of the following statements about the variance function is accurate?
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What is the relationship between $θ$ and the mean $µ$ in the given distributions?
What is the relationship between $θ$ and the mean $µ$ in the given distributions?
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What is the probability density function (PDF) for the Poisson distribution given in the table?
What is the probability density function (PDF) for the Poisson distribution given in the table?
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What does heteroscedasticity refer to in the context of residuals?
What does heteroscedasticity refer to in the context of residuals?
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In the context of the Gamma distribution, what does $v$ represent?
In the context of the Gamma distribution, what does $v$ represent?
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What can be inferred about the parameters $n$ and $v$ in the table?
What can be inferred about the parameters $n$ and $v$ in the table?
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How can outliers be addressed in the model adjustments?
How can outliers be addressed in the model adjustments?
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What adjustment can be made to address heteroscedasticity in a model?
What adjustment can be made to address heteroscedasticity in a model?
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Which of the following statements about the Deviance Residuals is correct?
Which of the following statements about the Deviance Residuals is correct?
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In the context of exponential dispersion families (EDF), what is the purpose of the parameter p?
In the context of exponential dispersion families (EDF), what is the purpose of the parameter p?
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Which member of the exponential dispersion family is most commonly recognized?
Which member of the exponential dispersion family is most commonly recognized?
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What does the fitted loss for the first observation in the table indicate?
What does the fitted loss for the first observation in the table indicate?
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What is the relevance of setting weights for observations outside the last n diagonals?
What is the relevance of setting weights for observations outside the last n diagonals?
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What does the dispersion parameter, $ heta$, imply when it is identical for all cells in the context of the information provided?
What does the dispersion parameter, $ heta$, imply when it is identical for all cells in the context of the information provided?
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What outcome is reached by correcting future incremental loss estimates for bias in the context provided?
What outcome is reached by correcting future incremental loss estimates for bias in the context provided?
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Which of the following statements about reserve estimates for the ODP Mack and ODP Cross-Classified models is true?
Which of the following statements about reserve estimates for the ODP Mack and ODP Cross-Classified models is true?
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When using the chain ladder method, how do you calculate the LDF for periods 1 to 2?
When using the chain ladder method, how do you calculate the LDF for periods 1 to 2?
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During the calculation of $eta_j$ parameters, which direction do you move in the table?
During the calculation of $eta_j$ parameters, which direction do you move in the table?
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Which of the following is NOT a step in calculating the incremental losses using the chain ladder method?
Which of the following is NOT a step in calculating the incremental losses using the chain ladder method?
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What is the significance of having a full triangle of data?
What is the significance of having a full triangle of data?
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What type of distribution is the EDF limited to under the given conditions?
What type of distribution is the EDF limited to under the given conditions?
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What is the formula used to calculate the loglikelihood for the saturated model?
What is the formula used to calculate the loglikelihood for the saturated model?
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In the calculation of the loglikelihood, what does the symbol ϕ represent in this context?
In the calculation of the loglikelihood, what does the symbol ϕ represent in this context?
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What is the result for the loglikelihood of the nested model when Yi = 120 and µ = 114.17?
What is the result for the loglikelihood of the nested model when Yi = 120 and µ = 114.17?
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What do the fitted losses (unsaturated) have in common across the different categories?
What do the fitted losses (unsaturated) have in common across the different categories?
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How is the loglikelihood of the saturated model calculated from the data?
How is the loglikelihood of the saturated model calculated from the data?
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What does the symbol D ∗ (Y, Ŷ ) represent in the context of deviance?
What does the symbol D ∗ (Y, Ŷ ) represent in the context of deviance?
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Which calculations were required to derive the incremental values of µ for the loglikelihood?
Which calculations were required to derive the incremental values of µ for the loglikelihood?
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How do the fitted losses for the saturated model compare when looking at the first category?
How do the fitted losses for the saturated model compare when looking at the first category?
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Study Notes
Advanced Reserving Methods
- This document discusses advanced methods for estimating loss reserves, particularly focusing on the chain ladder method and alternative distributions.
Chain Ladder Method
- The chain ladder method is a widely used stochastic method for reserving.
- It estimates loss development factors (LDFs)
- It uses incremental losses from each period
- The method assumes that losses develop over time in a predictable pattern.
- It calculates reserves using a system of multiplying incremental losses by these predictable factors.
Stochastic Models
- The chain ladder method yields the maximum likelihood estimate (MLE) of loss reserves.
- Additional models are presented to strengthen this outcome.
Non-Parametric Mack Model
- Accident years are independent.
- Losses follow a Markov Chain (dependent only on the previous period).
- Expected future cumulative losses are proportional to current losses.
- Variance of future losses is proportional to current losses.
- Conventional chain ladder estimates of LDFs are unbiased and have minimum variance.
- Conventional chain ladder estimates of reserves are unbiased.
Parametric (EDF) Mack Models
- Cumulative losses in the next period are distributed according to the exponential dispersion family (EDF).
- All assumptions of the non-parametric Mack model must be met.
- MLEs of loss development factors are equal to conventional chain ladder LDFs
- MLEs of LDFs are unbiased
- If the distribution is ODP and dispersion parameters are constant, then conventional LDFs are minimum variance unbiased estimators (MVUE)
Cross-Classified Models
- Incremental losses are independent.
- Losses have a distribution belonging to the exponential dispersion family.
- Expected incremental loss E[Ykj] = αkβj where Ykj is the incremental loss and Bj > 0. (Bj > 0)
- The sum of ßj = 1.
- Implied row parameters are in the LDF's row parameter section.
- MLE estimates of future losses and reserves are the same as conventional chain ladder.
- Reserve estimates, corrected for bias, are MVUEs.
Generalized Linear Models (GLMs)
- The chain ladder model and its extensions can be represented as GLMs.
- GLMs allow for estimation using statistical software, which provides more in-depth statistical information.
- The GLM represents reserves using a design matrix X and a parameter vector A.
- A relationship has been expressed between observations and covariates using a link function h().
Deviance
- This is used to assess goodness-of-fit for GLMs.
- It compares the log-likelihood of the fitted model to the saturated model (model with parameter for every observation).
- It can be calculated for different distributions.
Residuals
- Standardized Pearson Residual and Standardized Deviance Residuals are utilized to evaluate a model's quality of fit.
- Pearson Residuals: (actual – expected) / standard deviation
- Deviance Residual: The sign of (actual – expected) * (unscaled deviance for that component / estimated scale parameter)
Adjustments to the Model
- Heteroscedasticity (variance of residuals varies across periods).
- Outliers (extreme values).
- Restricted periods to only look at recent experience to prevent biases.
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Description
This quiz explores advanced methods for estimating loss reserves in insurance, with a particular emphasis on the chain ladder method and its stochastic modeling techniques. It covers the principles of loss development factors and alternative distributions to strengthen reserve estimation. Test your understanding of these essential actuarial concepts!
