Sahasrabuddhe Flashcards
Briefly describe the purpose of the Sahasrabuddhe method
It provides an approach for adjusting a development pattern for any claim layer to produce a development pattern for any other layer.
The approach allows for cost level adjustments and ensures that assumptions related to claim size models, claims development and trend are internally consistent.
Name 2 difficulties with the Pinto/Gingol approach used to estimate excess layer development.
- Access industry data may be difficult
- Methodology does not use the inherent relationship between claims size models, trend and claims development patterns.
Sahasrabuddhe’s method aims to address these issues.
Name 2 characteristics required to use development patterns from UNADJUSTED data and apply them to claims for all exposure periods (Sahasrabuddhe main method).
- When claims data are ground-up unlimited (GUU)
- When trend only acts in the AY direction
Since these 2 things are often not true in practice, we need to adjust the data before estimating development factors. Specifically, we need to adjust the data for differences in cost level and limits.
Calculate Exponential Claim Size Model LEV
LEV(x) = theta*(1-exp(-x/theta))
LEV(lim) = Trended mean*(1-exp(Lim/Trended Mean))
Name 2 requirements of the claim size model
- Claim size model parameters can be adjusted for the impact of inflation
- LEV and unlimited means can be easily calculated
Calculate the adjusted cumulative loss to a common cost level and Basic limits.
C’_AY,k = C_AY,k * LEV(B given theta_n,k)/LEV(L given theta_AY,k)
Adj Loss_AY,k) = Loss_AY,k * LEV(B_latest AY,k)/LEV(L_AY,k)
If B = L, data is just adjusted for trend in layer
Briefly describe the basic limit B
Limit where data is sufficiently credible to estimate development pattern.
Calculate development pattern for any layer and cost level
F^x_AY,k = F^B_n,k * LEV(X given theta_AY,inf)/LEV(B given theta_n,inf) / LEV(X given theta_AY,k)/LEV(B given theta_n,k)
CDF^x = CDF^basic * LEV(X_AY,ult)/LEV(B_latestAY,ult) / LEV(X_AY,k)/LEV(B_latestAY,k)
F represents the age-to-ult factors
Briefly explain how factors are primarily driven by the size of the claim size parameters
- Primary layers (layers that start at 0)
Larger diff in LEV(B) and LEV(Lim) as claim size increases. - Excess layers (layers that do not start at 0)
Larger diff in LEV between various limits as claim size decreases.
The diff between std chain-ladder factors and the adjusted factors will also increase with higher trend rates and longer development patterns.
Summarize the steps of the Sahasrabuddhe main method
- Calculate trend indices
- Develop claim size model parameters at the latest cost level
- Calculate trend-adjusted claim size model parameters using trend indices
- Calculate LEVs for the basic limit and the limit in the raw data triangle
- Calculate adjusted cumulative loss triangle at basic limit and latest cost level
- Calculate basic limit development pattern at the latest cost level
- Calculate development pattern for any desired layer and cost level based on the basic limit development pattern at the latest cost level
- Use adjusted development patterns to calculate ultimate losses and reserves for any desired layer and cost level
Briefly describe 2 practical issues with main method
- In practice, we are typically only provided with a development pattern (cost level not stated)
Solution: assume development pattern is at the latest cost level - We typically only have claim size models at ultimate (no claim size models at earlier age.
The implied model aims to solve these issues.
Calculate the development pattern for other layers and cost levels using the simplified model
Rj at ultimate:
R_AY,ult = LEV(X given theta_AY,inf)/LEV(B given theta_AY,inf)
Rj at age:
R^select_AY,k) = R_AY,ult + (1-R_AY,ult)*Decay_k
F^x_AY,k = F^B_n,k * LEV(X given theta_AY,inf)/LEV(B given theta_AY,inf) / R_k
Briefly describe decay model
Ensures that Rj is high at early maturities (closer to 1) and low at later maturities (further to 1). At ultimate, decay factor is 0 and Rj = U.
Note: a decay model only applies to primary layers.
Briefly describe the 3 properties of Rj
If the new layer, X, is lower than basic limit, then Rj smaller than 1 and we should see the following:
- Rj decreases as age increases
This is because there will be less development in the excess layer for earlier maturities. - Rj > U = lim Rj at ultimate, calculated as the ratio of LEVs at ultimate between layer X and B
Rj should keep decreasing since more development at older maturities will be above layer X. - If the base CDFs are from the ground-up unlimited layer, then max(Rj) = U*CDF
Describe the 3 Sahasrabuddhe method assumptions that are simple to implement
- Must select basic limit that is sufficiently credible for development factors.
When analyzing a loss triangle, actuaries make an implicit assumption that limit associated with the triangle is sufficiently credible. - Requires a claim size model
The claim size model is only used for relative LEVs, so the simple model may be OK if ratios of LEVs are reasonable. - Loss triangle must be adjusted to basic limits and a common cost level.
Given claim size and trend info, this is fairly simple to implement.
Describe the 2 Sahasrabuddhe method assumptions that could be burdensome (difficult to implement)
- Claim size models for maturities prior to ultimate are generally not available but necessary for the method.
- A table of trend indices is required for cumulative claim activity, but:
a) Trend typically applies to incremental claims
b) Impact of trend on reported incurred claims and the timing of trend’s effect on case reserves is difficult to ascertain.
True or False?
Given the development pattern limited factors, it is appropriate to use the pattern on all exposure periods.
False
Limited development patterns are a function of maturity AND cost level so they SHOULD NOT be used necessarily for all exposure periods.
Adjusting the pattern to different exposure periods may be immaterial when:
1. Development pattern is short
2. Trend rates are low
3. Limits are above the working layer
Briefly describe 2 modelling choices that would create large differences between modelled factors and unadjusted factors.
- Different AY (or CY) trend assumptions
- Different claim size model
Briefly explain how a larger theta for the claim size model results in deviated unadjusted development factors.
Larger theta means a higher % of losses will be capped at the limit and a larger difference in the LEVs between basic limit and policy limit of data in the triangle.
This causes further deviation because CDFs are adjusted using ratios of LEVs.
Explain how the estimate of the ultimate loss to the insurer would change if a smaller model parameter was used.
A smaller theta means smaller percentage of losses will be capped at the limit and there will be a smaller difference in the limited expected values between the various limits.
A smaller theta will produce an even smaller denominator (that outpaces the reduction in numerator on a % basis) creating a leveraged effect. This creates a larger LDF and larger ultimate losses for the insurer.
