Shapland Flashcards
Notation
w = AY from 1 to n
d = development period from 1
k = diagonal
c(w,d) = cumulative losses
q(w,d) = incremental losses
f(d) = factor applied to c(w,d) to estimate q(w,d+1)
F(d) = factor applied to c(w,d-1) to estimate c(w,d)
ODP model
Uses a GLM to model incremental claims using a log link function and an ODP distribution
Linear predictor = alpha_w + SUM(beta_d)
with alpha_1 > 0 and beta_1 = 0
Development Factors
F(d) = SUM(c(w,d)) / SUM(c(w,d-1))
Ultimate Claims
c(w,n) = c(w,d) * PROD(F(d))
Expected Value and Variance of Incremental Claims
E(q(w,d)) = m_w,d
Var(q(w,d)) = scale parameter * m_w,d^z
m_w,d = exp(linear predictor)
Z = 0 -> Normal
Z = 1 -> ODP
Z = 2 -> Gamma
Z = 3 -> Inverse Gaussian
Estimating the 𝛼 & 𝛽 Parameters Using a GLM
- Apply the log link function to the incremental loss triangle
- Specify the GLM using a system of equations with 𝛼 and 𝛽 parameters
- Convert to matrix notation 𝑌=𝑋𝐴, where 𝑌 is the matrix of log-incremental data, 𝑋 is the design matrix, and 𝐴 is the matrix of 𝛼 & 𝛽 parameters
- Use a numerical technique to solve for model parameters in the 𝐴 matrix to minimize he squared difference between the 𝑌 matrix and the solution matrix
ODP Bootstrap Model
The fitted incremental claims will exactly equal the standard CL method
- A simple link ratio algorithm can be used in place of the more complicated GLM algorithm, while still maintaining an underlying GLM framework
- The use of the age-to-age factors serves as a bridge to the deterministic framework; this allows the model to be more easily explained
- In general, the log link function does not work for negative incremental claims; using link ratios remedies this problem
Limitations:
- Does not adjust for CY effects
- Since it includes a parameter for each AY and development period beyond the first period, it might over-fit the data
Unscaled Pearson Residuals
r_w,d = (q(w,d) - m_w,d) / SQRT(m_w,d^z)
scale parameter = SUM(r_w,d^2) / (N - p)
N = # of data cells in the triangle
p = # of parameters = 2 * # AYs - 1
Distribution of the Residuals
Although sampling with replacement assumes the residuals are independent and identically distributed (i.i.d.), it does not require the residuals to be normally distributed
- This is an advantage of the ODP bootstrap model since the actual distributional form of the residuals will flow through the simulation process
- This is sometimes referred to as a “semi-parametric” bootstrap model since we are not parameterizing the residuals
Sample Triangles
Sampling with replacement from the residuals can be used to create new sample triangles of incremental claims
q*(w,d) = r * SQRT(m_w,d^z) + m_w,d
where r is a randomly select residual from the sample
The new sample triangles can then be cumulated, and age-to-age factors can be calculated and applied to calculate reserve point estimates and produce a distribution of point estimates
Process Variance in the Future Incremental Losses
The point estimates do not fully capture the predictive distribution of future losses
- incorporates process variance and parameter variance in the simulation of the historical data
- fails to incorporate process variance in the simulation of the future data
Rather than projecting each incremental loss using age-to-age factors, we sample from a Gamma distribution with mean m_w,d and variance scale parameter * m_w,d
- GAMMAINV(RAND(), alpha, beta)
- alpha = m_w,d / scale and beta = scale
Degrees of Freedom Adjustment Factor
Multiply the distribution of point estimates by the degrees of freedom adjustment factor and the scale parameter
f_DOF = SQRT(N/(N - p))
Allows for over-dispersion of the residuals in the sampling process and adds process variance to future incremental losses
Scaled Pearson Residuals
Another way to add over-dispersion in the residuals is by multiplying the unscaled Pearson residuals by the degrees of freedom adjustment factor
rs_w,d = r_w,d * f_DOF = r_w,d * SQRT(N/(N - p))
Standardized Residuals
rh_w,d = r_w,d * fh_w,d
fh_w,d = SQRT(1 / (1 - H_i,i))
where H_i,i is the ith point on the diagonal of the hat matrix
Considered as a replacement for and an improvement over the degrees of freedom adjustment factor
Standardized residuals are what should be sampled from when running the ODP bootstrap model
Approximating the Scale Parameter
Calculated using the unscaled Pearson residuals r_w,d
Can approximate using the standardized residuals rh_w,d
scale parameter = SUM(r_w,d^2) / N
Bootstrapping the Incurred Loss Triangle
- Run a paid data model in conjunction with an incurred model. Use the random payment pattern from each iteration of the paid data model to convert the ultimate values from each corresponding incurred model iteration to develop paid losses by AY.
- allows us to use the case reserves to help predict the ultimate losses, while still focusing on the payment stream for measuring risk - Apply the ODP bootstrap to the Munich Chain-Ladder (MCL) model. The MCL uses the inherent relationship/correlation between paid and incurred losses to predict ultimate losses. When paid losses are low relative to incurred losses, then future paid loss development tends to be higher than average.
- does not require us to model paid losses twice
- explicitly measures the correlation between paid and incurred losses
Bootstrapping the BF and Cape Cod Models
An issue with using the ODP bootstrap process is that iterations for the latest few accident years tend to be more variable than what we would expect given the simulations for earlier AYs
To address this issue, future incremental values can be extrapolated using the BF or Cape Cod method
GLM Bootstrap Model
Limitations:
- GLM must be solved for each iteration of the bootstrap model, which may slow down the simulation
- The model is no longer directly explainable to others using age-to-age factors
Benefits:
- We can specify fewer parameters, which helps avoid over-fitting
- We can add parameters for CY trends
- We can model shapes other than triangles
- We can match the model parameters to the statistical features found in the data
To calculate future incremental claims:
- Fit the same GLM model underlying the residuals to each sample triangle
- Use the resulting parameters to produce reserve point estimates
Negative Incremental Values–GLM Bootstrap Model
Negative incremental values can be problematic when parameterizing a GLM with a log link function since we cannot take the log of a negative number
- Modified Log-Link Function
- Used if the sum of incremental values in the column is positive
- For q(w,d) > 0, use ln(q(w,d))
- For q(w,d) = 0, use 0
- For q(w,d) < 0, use -ln[ABS(q(w,d))] - Subtracting the Largest Negative Value
- Used if the sum of incremental values in the column is negative
- Subtract the largest negative value from each incremental in the triangle
- Solve for the GLM parameters based on the logs of the adjusted incremental values
- Adjust the resulting fitted incremental values by adding the largest negative value
ODP doesn’t have this issue, but both models need to use modified unscaled Pearson residuals to deal with negative fitted values (take the absolute value under the square root)
Negative Values & Extreme Outcomes
Negative incremental values can cause extreme outcomes due to large age-to-age factors
- Identify the extreme iterations and remove them
- Recalibrate the model
- Limit incremental losses to zero
Missing Values
ODP:
- Estimate the missing value using surrounding values
- Exclude the missing value when calculating the loss development factors
- If the missing value lies on the last diagonal, we can either estimate the value or we can use the value in the second to last diagonal to construct the fitted triangle
GLM:
- The missing data simply reduces the number of observations used in the model
- Could use one of the methods described above to estimate the missing data if desired
Outliers in the Original Dataset
ODP:
- Exclude the outliers completely
- Exclude the outliers when calculating the age-to-age factors and the residuals (similar to missing values), but include the outlier cells during the sample triangle projection process (different from missing values)
GLM:
- Treated like missing data
- If the data is not considered representative of real variability, the outlier should be excluded, and the model should be parameterized without it
Heteroscedasticity
Non-constant variance of the residuals
This is an issue because the model assumes residuals are iid when we sample from the residuals
- Stratified Sampling
- Group development (or accident) periods with homogeneous variances and sample in each group separately
- Straightforward and easy to implement
- Some groups may only have a few residuals in them, which limits the amount of variability in the possible outcomes - Calculating Variance Parameters
- Group development (or accident) periods with homogeneous variances and calculate the standard deviation of each hetero group using standardized residuals
- Hetero adjustment factor for each group h_i = SD(all residuals) / SD(group i residuals)
- Multiply each residual by h_i so we can sample with replacement from among all residuals
- Divide the resampled residuals by h_i when calculating fitted values (i being the placement of the value in the new triangle, not the placement of the sampled residual) - Calculating Scale Parameters
- Group development (or accident) periods with homogeneous variances and calculate each hetero group’s scale parameter using the unscaled Pearson residuals
- Scale_i = N/(N-p) * SUM(residuals^2) / n_i
- Include the number of hetero groups - 1 as parameters
- h_i = SQRT(scale) / SQRT(scale_i)
- Multiply each standardized residual by h_i
- Divide the resampled residuals by h_i when calculating fitted values
Exposure Adjustment
ODP:
- Divide the claim data by earned exposures for each accident year
- The entire bootstrap process is run on the exposure-adjusted data
GLM:
- Model is fit to the exposure-adjusted losses
- The fit is exposure-weighted, meaning that exposure-adjusted losses with higher exposure are assumed to have a lower variance
- The exposure adjustment could allow fewer AY parameters to be used
Tail Factors
ODP:
- Instead of using a deterministic tail factor, we can assign a distribution (ex. normal or lognormal) to the tail factor parameter
GLM:
- Assume that the final development period will continue to apply incrementally until its effect on the future incremental claims is negligible
Fitting a Distribution to the Residuals
If we believe that extreme observations are not captured well in the loss triangle, we can parameterize a distribution for the residuals (such as normal) and resample using the distribution - know as parametric bootstrap
Residual Graphs
Can be used to test the assumption that residuals are i.i.d.
Should be able to draw a relatively horizontal line through the residuals
Should see constant spread (homoscedasticity)
Normality Test
Although we do not require the residuals to be normally distributed, it’s still helpful to compare residuals against a normal distribution - allows us to compare parameter sets and assess the skewness of the residuals, run before and after heteroscedasticity adjustments
Normality Plot:
- Shows the relationship between theoretical quantiles of a standard normal distribution (i.e., the 𝑥-axis) and the empirical quantiles of the observed data (i.e., the 𝑦-axis)
- If the data is normally distributed, then points will lie in a diagonal line
Test Values:
- 𝒑-value from Shapiro’s Test for Normality should be less than 5%
- R^2 should be close to 1
- AIC/BIC should be small, RSS comes from difference of residual and its normal counterpart from the normality plot
- AIC = 2p + n * [ln((2 * pi * RSS) / n) + 1]
- BIC = n * ln(RSS / n) + p * ln(n)
Outliers
Use box-whisker plots of the standardized residuals
When residuals are not normally distributed, outliers tend to be more common and we don’t necessarily want to remove them because they are representative of the shape of the data
How to find the optimal mix of parameters in the GLM bootstrap model
- Start with a “basic” GLM model which includes one parameter for accident, development, and calendar periods
- Check the residual plots for the basic model
- If the residuals by accident period, development period, and calendar period are not randomly scattered around zero, then we should consider adding parameters
- If certain parameters are not statistically significant, we should remove them
- The implied development pattern should look like a smoothed version of the CL development pattern - We keep adding/removing parameters until we see a proper residual plot
Estimated Unpaid Model Results
Standard Error:
- Should increase from older to more recent years because the standard error should follow the size of the increasing unpaid loss reserve
- Total standard error should be larger than the standard error for any individual year
- Total standard error should be less than the sum of the individual standard errors (but this can be true for unreasonable models)
CoV:
- Should generally decrease from older to more recent years because older years have a smaller loss reserve and there are few claims payments remaining
- Parameter uncertainty may overpower process uncertainty leading to an increased CV in most recent year
- Total CoV should be less than any individual year because the model assumes independent AYs
Run Models with the Same Random Variables
Each model is run with the exact same random variables (i.e.,random residuals in terms of position)
Once all of the models have been run, the incremental values for each model are weighted together for each iteration by AY
Causes correlation in model results since each model is run using the same set of random residuals
Run Models with Independent Random Variables
Each model is run with its own random variables (i.e.,different samples of random residuals in terms of position)
Once all of the models have been run, the weights are used to select a model for each iteration by AY
For example, suppose we are estimating unpaid losses using CL and BF bootstrap models. Further suppose the weights are 25% and 75%, respectively. For each iteration by AY, we would draw a uniform random variable on (0, 1). If the drawn uniform random variable is “<0.25”, then the CL unpaid losses would be used for that iteration/AY combination. Otherwise, the BF unpaid losses would be used. The result is a weighted mixture of models, where the CL and BF model results represent approximately 25% and 75% of the iterations by AY, respectively.
Location Mapping
For each iteration:
1. Sample residuals for segment 1
2. Track the location in the residual triangle where each sampled residual was taken
3. For all other segments, sample the residuals from their residuals triangles using the same locations
Advantages:
- Easily implemented
- Does not require an estimated correlation matrix
Disadvantages:
- Requires all business segments to have the same size data triangles with no missing data
- Since the correlation of the original residuals is used, we cannot test other correlation assumptions for stress testing purposes
Re-sorting
Re-sorting relies on algorithms such as Iman-Conover (rank correlation algorithm) or copulas to induce a desired correlation
For example, we could induce correlation among business segments by re-sorting the residuals until the rank correlation between each business segment matches the desired correlation specified by a correlation matrix
Advantages:
- Data triangles can be different shapes/sizes by segment
- Can use different correlation assumptions
- Different correlation assumptions may have other beneficial impacts on the aggregate distribution
Why modelling paid losses/claim counts may be preferable to modelling reported losses using an ODP
The variance of the ODP doesn’t allow for negative expected incremental losses (would give negative variance)
Paid losses and claim counts are less likely to show negative incremental development, which is more likely for reported losses
