Final Review Flashcards
Benktander
Formula and Pro/Cons
% reported * C-L Ult + % unreported * B-F Ult
Cum Losses + % unreported * B-F Ult
Advantages:
* Almost always has a smaller MSE than C-L and B-F Method
* Incorporates prior expectation of losses rather than fulyl relying on paid/reported losses to date (like the C-L)
Hurlimann
Ult, Credibility, ELR, p
Ult = Z * C-L Ult + (1-Z) B-F Ult
Credibility Z
* Z(GB) = p = % reported
* Z(WN) = p * ELR
* Z(OC) = p / [ p+sqrt(p) ] = p / ( p + t )
ELR
* m = incremental LR = sum(loss) / sum(premium)
* ELR = sum(m’s)
p = sum(m’s) / ELR for each AY
C-L = Individual | B-F = collective
Hurlimann
What Happens When Var(U) > Var(U_bc)?
- f increases
- t increases
- Z decreases
- more weight to B-F (collective) method
Brosius
What Happens When Intercept or Slope < 0?
Set intercept = 0 (C-L method)
Set slope = 0 (ELR method)
Brosius
Caseload Effect, Credibility Weighted Ultimate
All steps, another way to say VHM/EPV
% reported at 12 months = dy + x0
Z = VHM / (VHM + EPV)
VHM = Variation due to Loss Occurrence Process
EPV = Variation due to Loss Reporting Process
L(X) ult = Z * (x - x0)/d + (1-Z) * ELR Ult
USE ORIGINAL SCENARIO FOR ELR ULT
Brosius
Hugh White’s Question
3 Scenarios
Cov(X,Y) = Var(X) –> IBNR unchanged (B-F Method)
Cov(X,Y) < Var(X) –> decrease IBNR (ELR Method)
Cov(X,Y) > Var(X) –> increase IBNR (C-L Method)
Clark
Cape Cod Setup
Weibull
X = 6, 18, 30, etc
% Paid = G(X) = 1- exp( -(x/θ)^w )
Used-Up Prem = G(X) * OLEP
% Unpaid = 1 - G(X)
ELR = sum(Cum Loss) / sum(Used-Up Prem)
Ult = Cum Loss + % unreported * ELR * OLEP
Clark
Log-Likelihood
All steps/formulas
Expected Incremental Triangle = Incremental G(X) * ELR Ult
MLE Triangle = Actual * ln(Expected) - Expected
Log-Likelihood = sum(MLE Triangle)
Mack
Reserve Confidence Interval
All Formulas
Var = ln(1+ SE(reserves)^2 / reserves^2)
Confidence Interval = Reserves * exp(-Var/2 ± Z * SD)
Venter
Parameterized BF Method: h(w) / f(w)
What is it? + both formula
Variance is assumed to be constant
h(w) ~ kind of like ultimate loss, but ONLY to be used with f(d) to get expected reserves
f(d) ~ incremental % reported
If variance assumed to be constant:
* h(w) = sumprod[ f(d), incremental loss ROWS] / sumSQ[ f(d) ]
* f(d) = sumprod[ h(w), incremental loss COLUMNS] / sumSQ[ h(w) ]
w for the years, d for development period
Venter
Starting f(d) - seed
If not given in the problem:
* Incremental % Reported (first get % reported = 1/CDF)
* = incremental LDF / CDF
Venter
SSE and Adjusted SSE
SSE Triangle = (Actual - Expected)^2
SSE = sum(SSE Triangle EXCLUDING 1ST COLUMN)
Adjusted SSE = SSE / (n-p)^2
where n = # of values in triangle excluding the first column
p = number of parameters
C-L and CC: p = d - 1
BF: p = 2d - 2
where d = # of development periods
Venter
AIC / BIC
AIC = SSE * exp(2p/n)
BIC = SSE * n^(p/n)
where n = # of values in triangle excluding the first column
p = number of parameters
C-L and CC: p = d - 1
BF: p = 2d - 2
where d = # of development periods
Venter
Alternative Emergence Patterns
Linear with constant = f(d) * c(w,d) + g(d)
* Highly variable and slowly reporting lines (XS reinsurance)
Factor * Parameter = f(d) * h(w)
* Parameterized BF model
* Cape Cod has h(w) = h (constant)
Shapland
Unscaled Pearson Residual
= (actual - expected) / sqrt(expected^z)
USING INCREMENTAL LOSSES
The power, z, is used to specify the error distribution
* z = 0 for normal
* z = 1 for poisson
* z = 2 for gamma
* z = 3 for inverse gaussian
Shapland
Standardized Pearson Residual
What is this used for?
Unscaled Pearson Residual * hat matrix adj factor
where adj factor = sqrt (1 / (1 - hat matrix value) )
USED TO GET SAMPLED RESIDUALS (r) –> SAMPLED INCREMENTAL LOSS
= r * sqrt(fitted incremental loss) + fitted incremental loss
Shapland
Location Mapping
What is it + Pros/Cons
There exists correlation between LOBs.
- We would sample the residuals for the first LOB
- Make a note of that location of that sample
- Then sample the other LOBs from the same location
The residuals will be different from the LOBs (separate model, different parameters), but the sampled residual will come from the same location, which helps keep correlation consistent
Pro - Simple to implement in Excel and no need for correlation matrix
Con - Requires all LOBs to use data triangles of the same size with no missing values or outliers
Taylor
ODP Cross-Classified GLM Expected Incremental Losses
Setup + Calculation
Normalize α’s and β’s so that sum(β’s) = 1
α’s = Expected Ultimate Loss (column)
β’s = Expected Incremental % Reported (row)
Expected Incremental Loss = α’s * β’s
Meyers
K-S Test
K-S Test
1. Sort Actuals
2. Expected = 100 * (1/n, 2/n, etc)
3. abs difference
4. CV = 136/sqrt(n)
5. if max(abs diff) > CV, then model is invalidated
P-P Plot (SHEAVY TAIL WRONG)
* Graph expected (x-axis) vs actual (y-axis)
* S-shape = light tail (variance too low)
* Reverse S-shape = heavy tail (variance too high)
* U-shape = bias high (mean too high)
* Reverse U-shape = bias low (mean too low)
Verrall
Baysian BF Model Incremental Paid
What does higher β imply?
Incr Paid = Z * C-L incr paid + (1-Z) * B-F incr paid
Z = p / (βΦ + p)
C-L incr paid = (LDF - 1) * Cum Loss
B-F Incr paid = (Incr % Paid) * ELR Ult
ELR Ult (expert opinion) = M = α / β
Higher β’s means more certain in the expected losses, giving more weight to the B-F method
p = % paid, prior
Sahasrabudde
LEV Formula
Loss * [ 1-exp(-limit / loss) ]
where loss = claim size model
Sahasrabudde
Adjusted Cumulative Losses
Cumulative Loss (limited) * LEV(basic limit, last row, same column) / LEV (same limit, same same)
“Bring it to basic limit, latest year”
“But first need to back out current LEV, same same”
ONLY USE THIS TO CALCULATE PRE-ADJUSTED LDFS
DO NOT USE THIS TO CALCULATE IBNR OR ULTS
Sahasrabudde
Adjusted LDF (to ult) Formula
What do I apply these LDFs to?
Pre-Adjusted LDF (to ult) * numerator / demonimator
num = [ LEV(upper lim, same row, last column) - LEV (lower limit)] / LEV(basic lim last row, last column)
demon = [ LEV(upper lim, same same) - LEV (lower limit)] / LEV(basic lim last row, same column)
Trick - (SLLL) - SAME, LAST LAST LAST
Trick - (SSLS) - SAME, SAME, LAST, SAME
Apply these LDF (to ult) to YTD losses for specific layer
Siewert
Implied Development Method
Steps + Pros/Cons
- Calculate unlimited ult loss using C-L
- Calculated limited ult loss using C-L
- Excess ult loss = (1) - (2)
Pros
* Incorporates actual loss emergence
* Provides excess loss estimate (at early maturities) even when excess losses have not emerged
* Limited loss LDFs are more stable than excess LDFs
Con - not directly focusing on excess loss development (misplaced focus)
Friedland
Which Treaty Best Promotes Stability?
- Non-proportional reinsurance is intended to provide stability by protecting the risks insured by the ceding company’s losses above a limit
- Proportional reinsurance is intended to provide capacity and surplus relief to ceding
Friedland
Loss Portfolio Transfer
- Reinsurer assumes all or part of future loss payments on policies written in prior years
- AKA - ceding insurer cedes all or part of their reserves (for prior years) to the reinsurer
- Reinsurer usually takes over claims management
- Ceding insurer increases surplus (less liabilities)
Friedland
Adverse Development Cover
- Ceding company also cedes reserves to reinsurer, but only if they exceed a certain threshold
- Threshold usually a dollar amount or a multiple of current reserves
- No transfer of reserves, ceding insurer maintains claims management
Friedland
Difficulty Working With Reinsurer’s Loss Triangles
- There is greater volatility in the LDFs factors for reinsurance than with primary insurance
- Tail factors are often higher for reinsurer’s because losses in higher layers tend to take longer to develop (and because reinsurer has data lag)
Friedland
Variability of LDFs on Proportional vs Non-Proportional Reinsurance
- LDFs are more volatile for non-proportional reinsurance than proportional because
- Non-proportional reinsurance (usually) covers excess losses (above retention) and proportional covers losses from the group up
Teng and Perkins
Total Developed Loss n-th Adjustment, L(n)
Capped Loss n-th Adjustment, CL(n)
Expected Premium at n-th Adjustment, P(n)
Expected Incremental Premium at n-th Adjustment, P(n) - P(n-1)
Formula Method
L(1) = SP * ELR * %L(1)
CL(1) = L(1) * LCR(1)
P(1) = ( BP + CL(1) * LCF ) * TM
P(2) - P(1) = [ CL(2) - CL(1) ] * LCF * TM
%L(n) = % Losses emerged (cumulative)
Teng and Perkins
PDLD Formula Method
Formula Method
PDLD(1) = [ BP/SP * TM / (ELR * %L(1) ] + (LCR(1) * LCF * TM)
= (BP + CL(1) * LCF) * TM / L(1)
PDLD(2) = ILCR(2) * LCF * TM
Also:
* PDLD(1) = P(1) / L(1) = (BP + CL(1) * LCF) * TM / L(1)
* PDLD(2) = [ P(2) - P(1) ] / [ L(2) - L(1) ]
* ILCR(2) = [ LCR(2) * %L(2) - LCR(1) * %L(1) ] / [ %L(2) - %L(1) ]
* ILCR(2) = [ CL(2) - CL(1) ] / (L2 - L1)
%L(n) = % Losses emerged (cumulative)
Teng and Perkins
PDLD & CPDLD Empirical Method
PDLD(1) = Prem(0-27) / Loss(0-18)
PDLD(2) = Prem(28-39) / Loss(19-30)
PDLD(3) = Prem(40-51) / Loss(31-42)
Subsequent = 0
CPDLD(n) = sumproduct(PDLD, % Loss Emerged) / sum(% Loss Emerged)
Teng and Perkins
Premium Responsiveness of Teng & Perkin vs Fisk-Gibbon
Premium responsiveness decreases over time for Teng & Perkin
Premium responsiveness is constant over time for Fisk-Gibbon
Marshall
Risk Margin For Normal and Lognormal Distribution
Normal
* Z * CoV
* norminv(%-tile,0,CoV)
Lognormal
* Var = σ^2 = ln(1 + CoV^2)
* Risk margin = exp(-Var/2 + Z * SD) - 1
