Venter Factors Flashcards
Notation
c(w,d) = cumulative
q(w,d) = incremental
f(d) = factor applied to c(w,d) to estimate q(w,d+1)
F(d) = CDF
AY one -> w = 0
Dev period one -> d = 0
CL Assumptions
- Expected INCREMENTAL losses in the next development period are proportional to CUMULATIVE losses to date
- Losses are independent between AYs
- The variance of the incremental losses in the next development period is proportional to losses to date with proportionality constant that varies with age
If these assumptions hold, the CL method gives the minimum variance unbiased linear estimator of future loss emergence
Implication 1: Significance of Factors (Assumption 1)
Run a regression analysis on the volume weighted LDFs
In excel, LINEST(y,x,FALSE,TRUE)
- FALSE = no intercept
- TRUE = provide statistics
- index 1,1 = slope
- index 2,1 = SE
If the absolute value of each factor is greater than 2 * SE, they are significant
To pass the implication, most factors should be significant
Implication 2: Superiority to Alternative Emergence Patterns (Assumption 1)
Compare emergence patterns using goodness of fit tests
- Adjusted SSE
- AIC
- BIC
Adjusted SSE
SSE / (n - p)^2
where n is the number of predicted points and p is the number of parameters
n = # of cells in triangle EXCLUDING first column
p = # AYs - 1
AIC
SSE * exp(2p / n)
n = # of cells in triangle EXCLUDING first column
p = # AYs - 1
BIC
SSE * n^(p/n)
n = # of cells in triangle EXCLUDING first column
p = # AYs - 1
Alternative Emergence Pattern 1: Linear with Constant
E(q(w,d+1)|data) = f(d) * c(w,d) + g(d)
p = 2 * (# AYs - 1)
If only factor is significant, this is the CL method
If only constant is significant, this is the additive CL method
Including a constant term is often significant in the first development period or two, especially for highly variable and long-tail lines
Alternative Emergence Pattern 2: Factor Times Parameter
E(q(w,d+1)|data) = f(d) * h(w)
Parameterized BF model
The next period’s expected emerged loss is a lag factor f(d) times the expected ultimate loss amount h(w)
p = 2 * (AYs - 1)
Reduced parameter BF model
- We can reduce parameters by grouping AYs using apparent jumps in loss levels and fitting a single h parameter to each group
- We can assume that subsequent periods all have the same expected percentage development
- We can also fit a trend line through the BF ultimate loss parameters
Alternative Emergence Pattern 3: Cape Cod Method
E(q(w,d+1)|data) = f(d) * h
Special case of parameterized BF with a constant h(w)
p = # AYs - 1
Works better for loss ratios because a single target ultimate loss ratio makes more sense than a single target ultimate loss
Alternative Emergence Pattern 4: Additive Chain-Ladder Method
E(q(w,d+1)|data) = g(d)
Special case of linear with constant with a significant constant and insignificant factor
p = # AYs - 1
Produces same fitted values and statistics as CC method when using least squares
Iterative Method for Fitting a Parameterized BF Model (Constant Variance)
If the variances of the residuals are constant over the triangle, use the standard least squares procedure
Uses incremental losses
- Initialize the f(d)s using incremental lag factors
- Calculate CDFs to create cumulative % emergence and difference them to get the incrementals - Calculate h(w)
- h(w) = SUM(q(w,d) * f(d)) / SUM(f(d)^2) over d - Calculate the next iteration of f(d)
- f(d) = SUM(q(w,d) * h(w)) / SUM(h(w)^2) over w - Repeat until convergence of f(d)s and h(w)s
Iterative Method for Fitting a Parameterized BF Model (Variable Variance)
If the variances of the residuals are not constant over the triangle, use the weighted least squares procedure
Assume residual variances are proportional to f(d)^p * h(w)^q - regression weights are the reciprocal
If p = q = 1:
- h(w)^2 = SUM(q(w,d)^2 / f(d)) / SUM(f(d)) over d
- f(d)^2 = SUM(q(w,d)^2 / h(w)) / SUM(h(w)) over w
Iterative Method for Fitting a Cape Cod Model
Assuming residual variances are constant over the triangle
h = SUM(q(w,d) * f(d)) / SUM(f(d)^2) over w and d
f(d) = SUM(q(w,d) * h) / SUM(h^2) over w
Only need one iteration as h will not change
Adjusting 𝑓(𝑑) Values
Multiply each f(d) by 1/SUM(f(d)) and multiply h(w) by SUM(f(d))
This ensures the f(d)s sum up to 1 and can be interpreted as incremental emergence and h(w) as expected losses
Implication 3: Test of Linearity (Assumption 1)
Scatter plot of raw incremental residuals for a development period against the prior cumulative loss
If there are strings of positive and negative residuals in a row, then a non-linear process may be indicated
Implication 4: Tests of Stability (Assumption 1)
Assumption 1 uses the same development factor for all accident years, so we would expect that the appropriate development factor is stable over time
- Plot incremental residuals over AY
- if there are strings of positive and negative residuals in a row, then the development factors might not be stable and we may not want to use the same factors for all AYs - Plot a moving average of a specific age-to-age factor over AY
- if there are clear shifts over time, instability exists, and we may want to use a weighted average of the factors
When factors are unstable:
1. Use a weighted average of the available factors, with more weight going to recent years
2. Adjust the triangle for instability (ex - changes in settlement rates, use BS)
Implication 5: Correlation of Development Factors (Assumption 2)
Doesn’t require factors to be in adjacent columns, unlike Mack
Based on a table of development factors
For a pair of development factor columns:
1. Calculate the correlation coefficient r for any pair of development factor columns with at least 3 elements
2. Calculate the test statistic T = r * SQRT((n - 2) / (1 - r^2))
- n is the number of elements in in the column
- T is t distributed with n - 2 degrees of freedom
3. Interpret T
- Use T.INV(1 - alpha/2, n - 2) to obtain critical value
- If abs(T) > critical value, r is significant and we REJECT the null hypothesis that the development factors are independent
- If abs(T) < critical value, r is not significant and we FAIL to reject
Testing for Correlation Across the Triangle
Number of testable pairs m = (# of AYs - 3) C 2
Calculate an upper bound for the number of pairs of allowable significant correlations
- Pick a number of standard deviations (default 3.33)
- Bound = m * alpha + number * SQRT(m * alpha * (1 - alpha))
- Any more than this would imply significant correlation
Appropriate to test the triangle as a whole because
- it is more important to know whether correlation prevails globally than finding a small portion of the triangle that is correlated
- at a 10% level of significance, 10% of the pairs of columns could exhibit significant correlation by random chance
Implication 6: Significantly High or Low Diagonals (Assumption 2)
Multiplicative Diagonal Effects:
- If diagonal w + d = 7 is 10% higher than normal, q(w,d) = 1.1 * f(d) * h(w) for w + d = 7 and q(w,d) = f(d) * h(w) otherwise
Additive Diagonal Effects for CL Method:
- Reorganize the incremental triangle into columns and add dummy variables for the diagonals
- A regression is run on the reorganized data and if the coefficients for the dummy variables are significant, assume CY effects exist
Additive Diagonal Effects for Additive CL Method:
- Replace the cumulative losses in the design matrix with 1s
Diagonal Effects as a Measure of Inflation
E(q(w,d)|data) = f(d) * g(w + d)
To include AY effects as well: E(q(w,d)|data) = f(d) * h(w) * g(w + d)
