THE RESERVE UNCERTAINTIES IN THE CHAIN LADDER MODEL OF MACK REVISITED

2019 ◽  
Vol 49 (03) ◽  
pp. 787-821
Author(s):  
Alois Gisler
Keyword(s):  
Taylor Expansion ◽  
Distribution Free ◽  
First Order ◽  
Ladder Model ◽  
Run Off ◽  
Full Picture ◽  
Chain Ladder ◽  
One Year ◽  
The One

AbstractWe revisit the “full picture” of the claims development uncertainty in Mack’s (1993) distribution-free stochastic chain ladder model. We derive the uncertainty estimators in a new and easily understandable way, which is much simpler than the derivation found so far in the literature, and compare them with the well known estimators of Mack and of Merz–Wüthrich.Our uncertainty estimators of the one-year run-off risks are new and different to the Merz–Wüthrich formulas. But if we approximate our estimators by a first order Taylor expansion, we obtain equivalent but simpler formulas. As regards the ultimate run-off risk, we obtain the same formulas as Mack for single accident years and an equivalent but better interpretable formula for the total over all accident years.

scholarly journals One-Year and Ultimate Reserve Risk in Mack Chain Ladder Model

Risks ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 152
Author(s):  
Marcin Szatkowski ◽  
Łukasz Delong
Keyword(s):  
Simulation Study ◽  
Value At Risk ◽  
Risk Measure ◽  
Estimation Error ◽  
Individual Development ◽  
Emergence Pattern ◽  
Ladder Model ◽  
Chain Ladder ◽  
One Year ◽  
The One

We investigate the relation between one-year reserve risk and ultimate reserve risk in Mack Chain Ladder model in a simulation study. The first goal is to validate the so-called linear emergence pattern formula, which maps the ultimate loss to the one-year loss, in case when we measure the risks with Value-at-Risk. The second goal is to estimate the true emergence pattern of the ultimate loss, i.e., the conditional distribution of the one-year loss given the ultimate loss, from which we can properly derive a risk measure for the one-year horizon from the simulations of ultimate losses. Finally, our third goal is to test if classical actuarial distributions can be used for modelling of the outstanding loss from the ultimate and the one-year perspective. In our simulation study, we investigate several synthetic loss triangles with various duration of the claims development process, volatility, skewness, and distributional assumptions of the individual development factors. We quantify the reserve risks without and with the estimation error of the claims development factors.

scholarly journals AN INDUSTRY QUESTION: THE ULTIMATE AND ONE-YEAR RESERVING UNCERTAINTY FOR DIFFERENT NON-LIFE RESERVING METHODOLOGIES

2014 ◽  
Vol 44 (3) ◽  
pp. 495-499 ◽  
Author(s):  
Eric Dal Moro ◽  
Joseph Lo
Keyword(s):  
Closed Form ◽  
Internal Models ◽  
Cape Cod ◽  
Best Estimate ◽  
To Come ◽  
Chain Ladder ◽  
One Year ◽  
The One ◽  
Best Estimates

AbstractIn the industry, generally, reserving actuaries use a mix of reserving methods to derive their best estimates. On the basis of the best estimate, Solvency 2 requires the use of a one-year volatility of the reserves. When internal models are used, such one-year volatility has to be provided by the reserving actuaries. Due to the lack of closed-form formulas for the one-year volatility of Bornhuetter-Ferguson, Cape-Cod and Benktander-Hovinen, reserving actuaries have limited possibilities to estimate such volatility apart from scaling from tractable models, which are based on other reserving methods. However, such scaling is technically difficult to justify cleanly and awkward to interact with. The challenge described in this editorial is therefore to come up with similar models like those of Mack or Merz-Wüthrich for the chain ladder, but applicable to Bornhuetter-Ferguson, mix Chain-Ladder and Bornhuetter-Ferguson, potentially Cape-Cod and Benktander-Hovinen — and their mixtures.

scholarly journals The Prediction Error of Bornhuetter/Ferguson

2008 ◽  
Vol 38 (01) ◽  
pp. 87-103 ◽  
Author(s):  
Thomas Mack
Keyword(s):  
Stochastic Model ◽  
Prediction Error ◽  
The Other ◽  
Development Pattern ◽  
Initial Estimate ◽  
Numerical Example ◽  
Run Off ◽  
Chain Ladder ◽  
The One

Together with the Chain Ladder (CL) method, the Bornhuetter/Ferguson (BF) method is one of the most popular claims reserving methods. Whereas a formula for the prediction error of the CL method has been published already in 1993, there is still nothing equivalent available for the BF method. On the basis of the BF reserve formula, this paper develops a stochastic model for the BF method. From this model, a formula for the prediction error of the BF reserve estimate is derived. Moreover, the model gives important advice on how to estimate the parameters for the BF reserve formula. E.g. it turns out that the appropriate BF development pattern is different from the CL pattern. This is a nice add-on as it makes BF to a standalone reserving method which is fully independent from CL. The other parameter required for the BF reserve is the well-known initial estimate for the ultimate claims amount. Here the stochastic model clearly shows what has to be meant with ‘initial’. In order to apply the formula for the prediction error, the actuary must assess his uncertainty about both sets of parameters, about the development pattern and about the initial ultimate claims estimates. But for both, much guidance can be drawn from the estimates itself and from the run-off data given. Finally, a numerical example shows how the resulting prediction error compares to the one of the CL method.

scholarly journals The Prediction Error of Bornhuetter/Ferguson

2008 ◽  
Vol 38 (1) ◽  
pp. 87-103 ◽  
Author(s):  
Thomas Mack
Keyword(s):  
Stochastic Model ◽  
Prediction Error ◽  
The Other ◽  
Development Pattern ◽  
Initial Estimate ◽  
Numerical Example ◽  
Run Off ◽  
Chain Ladder ◽  
The One

Together with the Chain Ladder (CL) method, the Bornhuetter/Ferguson (BF) method is one of the most popular claims reserving methods. Whereas a formula for the prediction error of the CL method has been published already in 1993, there is still nothing equivalent available for the BF method. On the basis of the BF reserve formula, this paper develops a stochastic model for the BF method. From this model, a formula for the prediction error of the BF reserve estimate is derived.Moreover, the model gives important advice on how to estimate the parameters for the BF reserve formula. E.g. it turns out that the appropriate BF development pattern is different from the CL pattern. This is a nice add-on as it makes BF to a standalone reserving method which is fully independent from CL. The other parameter required for the BF reserve is the well-known initial estimate for the ultimate claims amount. Here the stochastic model clearly shows what has to be meant with ‘initial’.In order to apply the formula for the prediction error, the actuary must assess his uncertainty about both sets of parameters, about the development pattern and about the initial ultimate claims estimates. But for both, much guidance can be drawn from the estimates itself and from the run-off data given. Finally, a numerical example shows how the resulting prediction error compares to the one of the CL method.

scholarly journals A Bayesian Internal Model for Reserve Risk: An Extension of the Correlated Chain Ladder

Risks ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 125
Author(s):  
Carnevale Giulio Ercole ◽  
Clemente Gian Paolo
Keyword(s):  
Internal Model ◽  
Predictive Distribution ◽  
Solvency Ii ◽  
Risk Profile ◽  
Formula One ◽  
Life Insurer ◽  
Chain Ladder ◽  
One Year ◽  
The One ◽  
Traditional Approaches

The goal of this paper was to exploit the Bayesian approach and MCMC procedures to structure an internal model to quantify the reserve risk of a non-life insurer under Solvency II regulation. To this aim, we provide an extension of the Correlated Chain Ladder (CCL) model to the one-year time horizon. In this way, we obtain the predictive distribution of the next year obligations and we are able to assess a capital requirement compliant with Solvency II framework. Numerical results compare the one-year CCL with other traditional approaches, such as Re-Reserving and the Merz and Wüthrich formula. One-year CCL proves to be a legitimate alternative, providing values comparable with the more traditional approaches and more robust and accurate risk estimations, that embed external knowledge not present in the data and allow for a more precise and tailored representation of the risk profile of the insurer.

scholarly journals Recursive Credibility Formula for Chain Ladder Factors and the Claims Development Result

2009 ◽  
Vol 39 (1) ◽  
pp. 275-306 ◽  
Author(s):  
Hans Bühlmann ◽  
Massimo De Felice ◽  
Alois Gisler ◽  
Franco Moriconi ◽  
Mario V. Wüthrich
Keyword(s):  
Life Insurance ◽  
Solvency Ii ◽  
Mean Square ◽  
Conditional Mean ◽  
Loss Reserving ◽  
Chain Ladder ◽  
One Year ◽  
The One ◽  
Development Result ◽  
Recursive Representation

AbstractIn recent Solvency II considerations much effort has been put into the development of appropriate models for the study of the one-year loss reserving uncertainty in non-life insurance. In this article we derive formulas for the conditional mean square error of prediction of the one-year claims development result in the context of the Bayes chain ladder model studied in Gisler-Wüthrich. The key to these formulas is a recursive representation for the results obtained in Gisler-Wüthrich.

scholarly journals The Prediction Error of the Chain Ladder Method Applied to Correlated Run-off Triangles

2004 ◽  
Vol 34 (02) ◽  
pp. 399-423 ◽  
Author(s):  
Christian Braun
Keyword(s):  
Prediction Error ◽  
Distribution Free ◽  
Run Off ◽  
Chain Ladder Method ◽  
Chain Ladder

It is shown how the distribution-free method of Mack (1993) can be extended in order to estimate the prediction error of the Chain Ladder method for a portfolio of several correlated run-off triangles.

scholarly journals The Prediction Error of the Chain Ladder Method Applied to Correlated Run-off Triangles

2004 ◽  
Vol 34 (2) ◽  
pp. 399-423 ◽  
Author(s):  
Christian Braun
Keyword(s):  
Prediction Error ◽  
Distribution Free ◽  
Run Off ◽  
Chain Ladder Method ◽  
Chain Ladder

It is shown how the distribution-free method of Mack (1993) can be extended in order to estimate the prediction error of the Chain Ladder method for a portfolio of several correlated run-off triangles.

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