Sahasrabudde Flashcards
Pinto/Gogol Approach
Excess layer development is estimated by fitting observed development factors using industry data
- Access to industry data may be difficult
- Methodology does not use the inherent relationship between claims size models, trend and claims development patterns
Notation
L(d,p) = layer with lower bound d and upper bound p
C_i,j(L) = cumulative loss in layer L for AY i at age j
Trend Factors
- Refers to the annual change in cost level for a loss layer
- Tend not to vary between accident periods
- Trend that occurs in the development period or calendar period direction is often not considered
- Tend not to vary by loss layer
Trend Indices
Applied to cumulative ground up ultimate losses rather than an annual rate of change
Build tables of n x n AY and CY trend indices and multiply together to get the total trend indices
- AY should be the same in the rows
- CY should be the same in the diagonals
Claim Size Model
LEV(x) = theta * (1 - exp(-x/theta))
- Parameters can be adjusted for the impact of inflation
- Limited expected values and unlimited means can be easily calculated
Given parameters at one cost level, use the combined trend indices to calculate parameters at other cost levels
Developing the Basic Limit Loss Development Pattern
Basic limit B = limit in which we have sufficient credibility to determine stable/accurate development patterns
Adjust raw loss data to the basic limit at the latest exposure period
Adj Data at i,j = Raw Data at i,j * LEV(B|theta_n,j) / LEV(L|theta_i,j)
Calculating Loss Development Patterns by Layer & Cost Level
Once we have the basic limit age-to-ultimate factors at the latest exposure period, we can calculate development factors for any layer π and any exposure period
F_i,j(X) = F_n,j(B) * (LEV(X|theta_i,inf) / LEV(B|theta_n,inf)) / (LEV(X|theta_i,j) / LEV(B|theta_n,j))
where F represents age-to-ultimate factors
If the layer doesnβt begin at 0, take the difference in the LEVs in the numerators
Main Method
- Calculate trend indices
- Develop claim size model parameters at the latest cost level
- Calculate trend-adjusted claim size model parameters using the trend indices
- Calculate LEVs for the basic limit and the limit in the raw data triangle
- Calculate the adjusted cumulative loss triangle at the basic limit and latest cost level
- Calculate the basic limit development pattern at the latest cost level
- Calculate the development pattern for any desired layer and cost level based on the basic limit development pattern at the latest cost level
- Use the adjusted development patterns to calculate ultimate losses and reserves for any desired layer and cost level
Issues with Main Method
- In practice, we are typically only provided with a development pattern (i.e., cost level not stated) - assume that that the development pattern is at the latest cost level
- Typically only have claim size models at ultimate
Simplified Model
F_i,j(X) = F_n,j(B) * (LEV(X|theta_i,inf) / LEV(B|theta_i,inf)) / R_j(X,B) = F_n,j(B) * U(X,B) / R_j(X,B)
U(X,B) = LEV(X|theta_i,ult) / LEV(B|theta_i,ult) = Ratio of LEV of X over LEV of B at ultimate
R_j(X,B) = LEV(X|theta_i,j) / LEV(B|theta_i,j)
Since we are assuming claim size models are not available prior to ultimate, the R_j(X,B) term must be estimated
R_j(X,B)
Assume R_j(X,B) < 1, meaning B > X
- R_a > R_b, b > a
- At early maturities, there will be less development in the XS layer, resulting in R_j close to 1 - R_a >= U where U is the ultimate ratio
- There is more development associated with the denominator of π (losses in layer π΅) than the numerator of π (claims in layer π)
Violations occur if there is negative development or if we assume an XS layer may develop more quickly than the working layer
Estimating R_j(X,B)
Decay model
R_j(X,B) = U + (1 - U) * Decay Factor
Ensures that R_j is high at early maturities and decreases at later maturities
At ultimate, DF = 0 and R_j = U
Assumes the first AY is at ultimate
Only applied to primary layers
Simplified Model for XS Layers
F_i,j(X) = F_n,j(B) * (1 - U(X,B)) / (1 - R_j(X,B))
Model Assumptions
- The procedures require us to select a basic limit
- The procedure requires the use of a claim size model
- The procedure requires that the data triangle be adjusted to a basic limit and common cost level
- The procedure requires claim size models at maturities prior to ultimate (generally not available and have limited application)
- The procedure requires a triangle of trend indices (specify the cost level associated with cumulative claims, which is an issue because trend typically occurs on an incremental basis)
