Cane Fall 2005 meeting
Stochastic Reserving and Reserves Ranges
Emmanuel Bardis, FCAS, MAAA
Fall 2005
This document was designed for discussion purposes only.
It is incomplete, and not intended to be used, without the
accompanying oral presentation and discussion.
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Presentation components
 Deterministic vs. Stochastic methods and ASOP #36
 Various types of stochastic models
 Criteria for model selection
 Aggregate claim liabilities distributions
 Range of reasonable estimates and materiality
standards
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How actuaries currently handle
reserve ranges?
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Deterministic vs. Stochastic Methods
 Deterministic methods provide the best estimate of
claim liabilities
 Stochastic methods are more informative than
deterministic methods
 Produce a full distribution of possible outcomes in
addition to the best reserve estimate
 Provide basis for evaluating range of reasonable
estimates and RMADs
 Provide confidence levels of held reserves
 Consider the volatility of the reserves for each
individual line, together with the correlation of losses
across the various lines
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Why such a limited use of stochastic methods?
 A general lack of understanding of the methods
 Lack of flexibility of the methods, including lack of
suitable software
 Lack of immediate need -- why bother when traditional
methods suffice for the calculation of a best estimate?
 Lack of a clear guidance (accounting or actuarial) on
how to calculate reserve estimates
 SSAP # 55 accounting guidance says that the held
reserves must be “management’s best estimate” (?)
of the actual liabilities
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Guidance according the ASOP #36
 According to ASOP #36 “Statement of Actuarial
Opinion Regarding Property/Casualty Loss and LAE
Reserves”
 “In estimating the reasonableness of the reserves,
the actuary should consider one or more expected
value estimates of the reserves, except when such
estimates cannot be made based on available data
and reasonable assumptions”
 Translation: We must find different ways to
calculate the expected value (i.e. the mean) of
the unknown distribution. The chance of getting
an expected value equal to the actual claim
liability amount is virtually zero.
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More ASOP #36…
 “Other statistical values such as the mode or the
median (50th percentile) may not be appropriate
measures for evaluating loss and LAE reserves, such
as when the expected value estimates can be
significantly greater than these other estimates”
 Translation: That’s easy! Actuaries must be
conservative
 “The actuary must use various methods to arrive at
expected estimates… it is not necessary to estimate or
determine the range of all possible values”
 Translation: We are, in essence, asked to
concentrate our efforts on calculating the statistical
mean, not the associated distribution
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ASOP#36 concludes
 “A range of reasonable estimates… could be produced
by appropriate actuarial methods… The actuary may
include risk margins in a range of reasonableness
estimates… a range… however, usually does not
represent the range of all possible outcomes”
 Translation: Look at various distributions and select
among them. ASOP here is unclear on
distinguishing between:
 The range of Best Estimates and
 The range of actual outcomes
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Calculation of ranges employing multiple
projection methods
The actual distribution has a wider range
Best estimate
Method #1
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Best estimate
Method #2
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Stochastic Theory and Various Types of
Models
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All risks that contribute to the uncertainty of
claim liabilities estimates
Total Risk
Process Risk
i.e. fair die
Actual claim
liabilities
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Parameter Risk
i.e. unfair die
Expected claim
liabilities
Model Risk
i.e. 6 unfair dice
Model Estimate
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”Chain Ladder” type of models
 Example: Mack model assumptions:
I. The expected loss amount at time n, is equal to the
product of the known paid loss amount through time (n-1)
times a “true” unknown loss development factor
II. No correlation among accident years exists
III. The variability of the link ratios is inversely proportional
to the magnitude of the loss amounts
 The Mack model provides the Mean and “Standard error”
of the claim liabilities, where:
 Standard error = E[(Actual–Model Estimate)2 | Triangle]
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“Simulation” type of models
 Bootstrapping is a powerful, yet simple, technique that
employs simulations, avoiding the fit of complex models
 The chain ladder method (CLM) produces identical
reserve estimates to a Generalized Linear Model model
(OPD)
 Incremental fitted payments from the CLM are
compared to historical increments, producing residuals
 Regression residuals are approximately independent
and identically distributed around zero
 Bootstrapping involves sampling with replacement of
the residuals. The simulated residuals produce
forecasted incremental payments
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“Incremental” type of models
 These modes employ log-incremental rather than
cumulative payments
 They fit curves across accident, development and
payment years producing “parsimonious” models
 The statistical framework allows the user of the model to
test the significance of the parameters
 Goodness of Fit tests allow the user to “tailor” the model
parameters to fit the characteristics of the data
 Examples are the Christofides “log-incremental” model
and the “ICRF’s” model
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Graphs for Incremental models
14,000
Cumulative P ayments
Standardized Residuals
Amount P aid
12,000
10,000
3
8,000
AY0
AY1
6,000
AY2
AY3
AY4
AY5
2
6
1
4,000
2,000
AY6
0
10.00
1
2
3
4
Development Year
5
Residuals
0
Log - I ncremental P ayments
7.00
6.00
5.00
4.00
AY0
AY1
AY2
AY3
1
2
3
4
5
6
AY4
AY5
-2
-3
AY6
3.00
0
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0
-1
8.00
P aid
Log of Amount
9.00
0
1
2
3
4
Development Year
5
6
Development Year
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Bayesian type of models
 Incremental models are fitted to historical data and
produce “best fitted” parameters. Future payments are
then calculated based on these parameters
 Bayesian models assume instead a “prior” distribution
of the parameters
 Based on the Bayes Theorem and the historical data,
a “posterior” distribution of the parameters is produced
 Monte Carlo simulations produce the distribution of the
parameters and the future payments
 Bayesian models incorporate the informed judgment of
the model’s users
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Criteria for model selection
 Data: Are the model assumptions satisfied by the data?
 Do correlation across accident years exist?
 Is there any negative loss development present?
 Cost/Benefit considerations:
 What is the marginal benefit of complicated models?
 What is the cost of “specialist” software?
 How difficult is to explain to management?
 Reasonability checks:
 Standard errors should increase for immature years
 Goodness of Fit: find a model that best “fits” the data
 Complexity must be appropriately penalized
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Aggregate Claim Liabilities Distribution
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What about aggregate distributions?
 The 75th percentile of the combined distribution of two
lines of business is generally NOT the sum of the 75th
percentiles of the individual distributions
 The former is true only in the case of perfect
correlation between the two lines. This is very
unlikely!
 Generally the aggregate distribution is less risky
than the sum of the risk of the parts
 The volatility of the aggregate distribution increases:
 By the volatility of the individual lines
 The correlation between the lines
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Impact to the 75th percentile of the aggregate
distribution
Correlation Impact for Two N(100,30) Lines
Impact of Correlation and Volatility
7.0%
20.0%
St Dev 25
18.0%
Increase over Uncorrelated 75th Percentile
75th percentile
5.0%
4.0%
3.0%
2.0%
1.0%
St Dev 50
St Dev 75
16.0%
St Dev 100
St Dev 200
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
Correlation Between Lines
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100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0.0%
0%
Increase over Uncorrelated 75th Percentile
6.0%
St Dev 10
0.0%
0.0%
25.0%
50.0%
75.0%
100.0%
Correlation Between Lines
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Theory of Copulas
 Copulas provide a convenient way to express the
aggregate distributions of several random variables
 Copula components:
 The distributions of individual random variables
 Correlations of these variables
 Correlation coefficients measure the overall strength of
association across various distributions
 Copulas can vary that degree of association over the
various parts of the aggregate distribution
 Example: for workers comp and property losses the
correlation is higher in the tail of the distribution
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Comparison of Copulas
925,000
Normal Copula
875,000
Student t
Loss amounts
825,000
Gumbel Copula
775,000
725,000
675,000
625,000
0.
99
0.
9
0.
8
0.
7
0.
6
0.
5
0.
4
0.
3
0.
2
0.
1
0.
01
575,000
Percentiles
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Range of Reasonable Estimates and
Standards of Materiality
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“Standards” for comparing actuarial estimates
 “Statistical” materiality in an actuarial/statistical context:
 Is the difference between two actuarial estimates
significant different from each other?
 Parameter risk is relevant here. We are concerned with
the variability of the “Expected” claim liabilities only
 “Financial” materiality in a financial reporting context:
 Would users of financial statements draw different
conclusions if a different reserve estimate is booked in
the financial statement?
 Total risk is relevant here. We are concerned with the
variability of the “Actual” claim liabilities
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We follow a “null hypothesis” testing approach
Type I error
Type I error
Significance level
Significance
level
Lower
Materiality
standard
Upper
Materiality
standard
Carried
reserves
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“Statistical” materiality standards
at a 7.5% significance level
 The coefficient of variation
(CV) of claim liabilities is
proportional to the “inherent”
risk of a line of business
Sta n d a rd s a s a % o f c a rrie d re se rve s
25.0%
Up p e r
L o we r
20.0%
 The range implied by the
statistical standards is
proportional to the volatility of
the line of business
CV
15.0%
10.0%
 “Financial” materiality
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5.0%
Liability
Other
Comp
Workers
Liability
Auto
0.0%
Personal
standards should consider the
risk of the claim liabilities in
comparison to
 Surplus and
 Net Income
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A little diversion:
Financial vs Statistical standards
 Even if the carried and indicated reserves are “statistically”
indistinguishable from one another the risk of material adverse
deviation still remains!
 It happens in the case where:
Upper “statistical “ range - carried reserves >=
Financial materiality standard
The Upper statistical range is produced by “appropriate” actuarial
methods and sets of assumptions…

Financial
Materiality
Lower
Statistical
range
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Carried
reserves
Standard +
Carried
reserves
Upper
Statistical
range
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