GLOBAL ACTUARIAL SERVICES PRACTICE
Practical Applications of the MACK and
BOOTSTRAPPING Methods When Estimating
Reserve Ranges
A D V I S O R Y
Scott P. Weinstein, FCAS, MAAA
Ash Ruparelia
2007 Casualty Loss Reserve Seminar
September 10, 2007

Presentation Overview
Understanding the Issues
Uncertainty – Where Does It
Come from?
Stochastic Approaches
Industry-Based Examples
Technical Walkthrough
Practical Considerations
Diagnostic Graphs
Q&A
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Understanding the Issues
For non-life insurance companies, loss reserves (technical reserves) comprise the majority of their liabilities
Uncertainty involved in estimating liabilities pose considerable risk
Adverse reserve run-off has led to insurance company downfalls
Negative impact on shareholders, policyholders, employees and other insurers
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Understanding the Issues
The nature or extent of this uncertainty is generally not well understood by decision-makers
Financial statement reporting requires that a single number represent the technical reserve
Potential investors and regulators generally recognize that the number may change over time
The magnitude of potential variation is generally not identified or quantified in an informative way
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Understanding the Issues
Why is the quantification of uncertainty important? A number of internal and external pressures:
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Understanding the Issues
Compliance Pressures
SEC increasingly requesting disclosures regarding reserve uncertainty, including potential financial statement impact
Australian Prudential Regulatory Authority already requires that technical reserves be determined as the present value of a central estimate, with risk margin to approximate the 75% confidence level
Impending regulatory guidance of International Financial Reporting
Standard’s (IFRS) Phase II requires the present value of a central estimate with an explicit additional margin for bearing risk
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Understanding the Issues
Compliance Pressures
Rating Agency Interest
Enterprise Risk Management
Capital Adequacy
Europe’s Solvency II Initiative
Market value of reserves based on expected present value of future cash flows, but will include a market value margin that meets the objectives of either third-party portfolio transfer or recapitalizing the company to ensure a proper run-off scenario
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Understanding the Issues
Implications for Senior Management, Boards of Directors and
Other Interested Parties
Economic Capital – a mechanism by which companies are measuring the risks of their business
Strong Risk Governance – allows for better recognition of the uncertainties inherent in claims liabilities. Enables informed decision-making and enhanced transparency with internal and external audiences
Merger or Acquisition Benefits
Reinsurance Strategy – Risk assumption and mitigation
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Uncertainty – Where Does It Come from?
Random nature of claims
Exposure to claims is uncertain
Number, size and timing of claims
Data
Homogeneity of data
Credibility of data
Other uncertainty
Model error – is the model reliable?
Change in underlying exposure over time
External influences / internal operational changes
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Uncertainty – Where Does It Come from?
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
DF’s
1 2 3 4 5 6 7
Uncertainty from estimation
8 9 10 11 12
Uncertainty from future process
Ult
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Uncertainty – Where Does It Come from?
Claim triangles encompass uncertainty from the past
The variability of the estimated claim around the “true” value of the distribution we are trying to measure indicates the estimation error
BUT, the future payments will also have variance i.e. the projected value in each future cell depends on the possible future outcomes
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Uncertainty – Where Does It Come from?
Therefore, must allow for the future variability – this is called process error
The prediction error measures the variability of the deviation C – Ĉ.
It can be shown that (approximately):
Prediction variance = estimation variance + process variance
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Stochastic Approaches
Deterministic actuarial techniques e.g. chain ladder
Ignores the random nature of the claims process
Stochastic approach
Statistical model to describe the assumptions of the underlying claim settlement process
Allows for the random nature of the claims process
Can test fit of model
Appreciate: the potential variability in the reserves the shape of the reserve distribution
Provides insight into the risk profile of the underlying business
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Stochastic Approaches
The distribution produces a range of possible outcomes, not the range of reasonable outcomes
Management must interpret the results in light of the intended purposes of the modeling
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Stochastic Approaches
Several stochastic reserving methods are gaining ground:
Mack
Bootstrap
Factorial
Generalized linear or other statistical models
Focus on results of Mack and Bootstrapping for the remainder of the presentation
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Stochastic Approaches
Mach Method - Details
It is a statistical model underlying pure chain ladder
It has three explicit assumptions
The expected value of cumulative claims at development period, k, is equal to cumulative claims at k-1 multiplied by the development factor, E[C i,k+1
| C i,1
....... C i,k
] = C i,k
.f
k
The cumulative claim amounts are independent between origin years for all development periods, i.e. {C i,1 i≠j are independent.
....... C i,n
}, {C j,1
....... C j,n
},
The variance of the cumulative claims at development period k, is proportional to the cumulative claims at k, i.e. Var[C i,k+1
C i,k
] = C i,k
.
σ 2 k
| C i,1
.......
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Stochastic Approaches
Mack Method - Details
A major consequence of the first and second assumptions is that the estimates of the development factors are unbiased and uncorrelated
Also, the estimates of the ultimate claims and the reserves are unbiased
The estimates of the development factors are such that they are minimum variance estimators amongst all linear estimates of the development factors
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Stochastic Approaches
Mach Method - Strengths
Stochastic chain ladder model
Easy to explain assumptions underlying model
No distribution assumption required
Can be applied to paid and incurred claims data
Sensible progression of standard error relative to mean reserve over the origin years
The development factors are minimum variance unbiased estimators of the true development factors
Can adjust model to exclude or weight specific development factors
Can incorporate a tail factor into the standard error calculation
(Mack 1999)
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Stochastic Approaches
Mack Method - Weaknesses
Distribution free – no automatic reserve ranges
Need to assume a distribution to produce ranges
Needs reasonable history to work sensibly – 10 years worth
History not necessarily a good guide to future
Assumptions may not be met in practice:
Non-independent origin years
Development factors correlated
Calendar year trends
Tail factor – left up to individual actuary’s judgement both to estimate the factor and the error components
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Stochastic Approaches
Bootstrapping - Details
Basic principle is to create many pseudo data sets from actual data set
Relies on having sufficient observations in the data otherwise create overlaps in pseudo data
Recently popular as computing power and storage has improved
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Stochastic Approaches
Bootstrapping - Details
Started as an ad hoc method of deriving variability and distribution of the reserves using the chain ladder method applied to past data
Ignored the modeling of the underlying claims process
Therefore the variability of the reserves was underestimated
Recent statistical models replicate chain ladder e.g.
Over-dispersed Poisson model, Negative Binomial and
Normal approximation
Bootstrap can be used with these statistical models to produce a fuller picture of the variance and distribution
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Stochastic Approaches
Bootstrapping - Strengths
Easy to set up in a spreadsheet - does not use complex formula or specialist software
More than point estimate - Can derive the variance and the simulated distribution of reserve outcomes
Tail variability can be allowed for in a pragmatic way e.g. by fitting a tail to the pseudo data using curve fitting
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Stochastic Approaches
Bootstrapping - Weaknesses
Over-dispersed Poisson model has constraints
Needs positive total development of claims for each development year
Will only model positive claim amounts
So, is not usually suitable for incurred claims where there are savings on case estimates
Need to simulate future payments according to model to obtain simulated distribution otherwise assume a distribution to produce ranges (if not simulating future claims increments)
Needs reasonable history to work sensibly – 10 years worth
History not necessarily a good guide to future
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Industry-Based Examples
Applied Mack and Bootstrapping to industry-based homeowners, commercial auto (motor) and workers’ compensation claims payment data
Due to credibility gained by examining aggregated industry data, the claims process appears to be fairly stable
The central reserve estimate of each method are reasonably consistent and the errors or standard deviations, relative to the reserves, are quite low
The increase in percentage error as the accident year matures is due to the smaller volume of claims still open in older time periods
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Industry-Based Examples
Chart 3 - Homeowners U.S. Industry Stochastic Outputs
Accident
Year Ultimate
1996 21,430,948
1997 16,709,712
1998 20,520,842
1999 21,094,145
2000 25,059,680
2001 28,805,081
2002 25,475,597
2003 27,033,384
2004 30,170,646
2005 33,943,504
Total 250,243,539
*Note: 1,000 Simulations
Mack Method
Reserve Std Deviation % Error
0.0%
12,268
32,918
57,170
131,283
7,077
11,254
13,419
27,412
57.7%
34.2%
23.5%
20.9%
298,606
528,242
1,073,903
2,348,894
9,933,875
14,417,159
45,545
54,964
88,958
168,281
857,819
899,435
15.3%
10.4%
8.3%
7.2%
8.6%
6.2%
Ultimate
21,430,948
16,709,755
20,520,330
21,094,082
25,059,022
28,802,744
25,472,828
27,028,717
30,166,699
33,953,188
250,238,312
Bootstrap Method *
Reserve Std Deviation % Error
0.0%
12,311
32,406
57,107
130,625
22,016
35,850
45,577
67,648
178.8%
110.6%
79.8%
51.8%
296,269
525,473
1,069,236
2,344,947
9,943,559
14,411,932
99,880
128,090
181,236
268,655
598,599
791,197
33.7%
24.4%
17.0%
11.5%
6.0%
5.5%
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Industry-Based Examples
Chart 4 - Commercial Motor U.S. Industry Stochastic Outputs
Accident
Year
1996
Ultimate
9,699,471
1997 10,197,376
1998 10,473,152
1999 11,094,052
2000 11,347,414
2001 10,921,222
2002 10,365,177
2003 10,195,575
2004 10,293,768
2005 10,573,375
Total 105,160,583
*Note: 1,000 Simulations
Mack Method
Reserve Std Deviation % Error Ultimate
28,952 0.0% 9,670,519
65,918 17,087 25.9% 10,166,995
141,606
242,995
480,722
24,662
32,499
38,741
17.4% 10,441,996
13.4% 11,061,234
8.1% 11,314,429
907,954
1,722,918
3,165,720
5,351,220
8,138,580
20,246,586
88,521
96,926
136,267
210,013
295,053
480,472
9.7%
5.6%
4.3%
10,889,171
10,333,779
10,166,055
3.9%
3.6%
10,257,799
10,559,285
2.4% 104,861,262
Bootstrap Method*
Reserve Std Deviation % Error
0.0%
35,537 17,074 48.0%
110,450
210,177
447,737
28,236
37,808
53,784
25.6%
18.0%
12.0%
875,903
1,691,520
3,136,200
5,315,251
8,124,490
19,947,265
73,591
103,310
149,937
222,383
350,439
521,682
8.4%
6.1%
4.8%
4.2%
4.3%
2.6%
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Industry-Based Examples
Chart 6 - Commercial Motor: Mack Paid - Normal Distribution of Reserves
$20,636,003
$20,246,586 $21,036,892
95th percentile
Mean
Selected Ultimate
18,500,000 19,000,000 19,500,000 20,000,000 20,500,000
($000's)
21,000,000 21,500,000 22,000,000
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Industry-Based Examples
Chart 7 - Homeowners: Mack Paid - Normal Distribution of Reserves
120%
100%
80%
60%
40%
20%
0%
11,500,000
95th Percentile
Mean
Selected Ultimate
12,500,000
$13,673,620 $14,417,159
13,500,000 14,500,000 15,500,000
$15,896,597
16,500,000
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Industry-Based Examples
The preceeding graphs showcase the possible outcomes around a statistical mean assuming a specifically modelled distribution
This result is not the same as a range of reasonable low and high technical reserve estimates
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Industry-Based Examples
Depending on the nature of the claims to which the company is exposed, the central estimate would likely fall somewhere near the statistical mean of the distribution
If the reserve estimation process occurs separately from the quantification of uncertainty, inconsistencies between the central estimate and the statistical mean could be the unintended consequences
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Analysis and Interpretation of the Output
Analysis of model outputs
Deriving ranges from model output
Interpretation
Interplay between best estimates and ranges
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Practical Considerations
Paid or incurred triangle?
Projection using the Incurred triangle Projection using the paid triangle
50,000,000 60,000,000 70,000,000 80,000,000 90,000,000 100,000,000 110,000,000 120,000,000 130,000,000 140,000,000 50,000,000 60,000,000 70,000,000 80,000,000 90,000,000 100,000,000 110,000,000 120,000,000 130,000,000 140,000,000
LogNormal 25% 75% Mean Best Estimate LogNormal 25% 75% Mean Best Estimate
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Practical Considerations
Prediction error can only reflect estimation error and statistical (process) error, BUT NOT SPECIFICATION
ERROR…
Model chosen can be wrong – chain ladder may not work well in cases where incremental payments are not dependent on previous cumulative payments.
When model assumptions are violated, prediction error may be significantly underestimated or be simply invalid
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Practical Considerations
If persistent trends are identified through any of the diagnostic graphs, the user may wish to adjust the actual data set to remove outliers (e.g. individual data points, entire years of origin, entire diagonals) to remove the biases
But removal of data may dampen the variability of the remaining dataset. Hence it is important to investigate the reasons for any observed bias.
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Practical Considerations
Other considerations:
Tail
Large losses
Diversification
Changes in exposure
Reinsurance
Converting from UY to AY
White noise
Hindsight testing
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Standardized Residuals by Origin
1. Scatter plot of residuals:
[(actual incremental – expected incremental) / square root of expected incremental]
2. Each column represents the residuals relative to the selected development factor for a given accident year
3. Red line is the average trend line
4. Ideally, the average trend line should center around zero, with small random negative and positive fluctuations
This column represents errors for accident year 1996
This column represents errors for accident year 2000
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Standardized Residuals by Development
1. Each column represents the residuals relative to the selected development factor at each maturity
2. Ideally, the average trend line should center very closely around zero, especially if the selected development pattern is based on an all-year average
This column represents residuals for maturity 1
This column represents residuals for accident maturity 5
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Standardized Residuals by Origin
1. Each column represents the residuals relative to the selected development factor by diagonal
2. Trend line represents calendar year trend
3. Ideally, the average trend line should center around zero, with small random negative and positive fluctuations
This column represents residuals for calendar year 2001
This column represents residuals for calendar 2005
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Presenter’s contact details
Scott Weinstein sweinstein@kpmg.com
KPMG LLP (US)
404 222 3594
Asheet Ruparelia ash.ruparelia@kpmg.co.uk
KPMG LLP (UK)
+44 (0)20 7694 2244
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©2007 KPMG LLP, a U.S. limited liability partnership and a member firm of the KPMG network of independent member firms affiliated with KPMG International, a Swiss cooperative.
All rights reserved.
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