estimation and application of ranges of reasonable estimates charles l. mcclenahan, fcas, maaa
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2003 Casualty Loss Reserve Seminar. Estimation and Application of Ranges of Reasonable Estimates Charles L. McClenahan, FCAS, MAAA. Introduction. “Range of Reasonable Estimates” Recent Development Once was informal ± 5% 5% of what was flexible - PowerPoint PPT PresentationTRANSCRIPT
Estimation and Application of Ranges of Reasonable Estimates
Charles L. McClenahan, FCAS, MAAA
2003 Casualty Loss Reserve Seminar
Introduction
“Range of Reasonable Estimates”
– Recent Development
– Once was informal ± 5%
- 5% of what was flexible
– 1973 Robert Anker review described three ranges
- Absolute Range = Lowest indication to Highest indication
- Likely Range = Lowest selected to Highest Selected
- Best Estimate Range
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Introduction (continued)
1988 Statement of Principles
– Principle 3 – “The uncertainty inherent in the estimation of required provisions for unpaid losses or loss adjustment expenses implies that a range of reserves can be actuarially sound. The true value of the liability for losses or loss adjustment expenses at any accounting date can only be known when all attendant claims have been settled.”
– Principle 4 – “The most appropriate reserve within a range of actuarially sound estimates depends on both the relative likelihood of estimates within the range and the financial context in which the reserve will be presented.”
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Introduction (continued)
AAA Committee on Property and Liability Financial Reporting
– “a reserve makes a ‘reasonable provision’ if it is within the range of reasonable estimates of the actual outstanding loss and loss adjustment expense obligations.”
– the “range of reasonable estimates is a range of estimates that would be produced by alternative sets of assumptions that the actuary judges to be reasonable, considering all information reviewed by the actuary.”
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Introduction (continued)
Actuarial Standards Board – ASOP No. 36 – Statements of Actuarial Opinion Regarding Property/Casualty Loss and Loss Adjustment Expense Reserves
– range of reasonable estimates is “a range of estimates that could be produced by appropriate actuarial methods or alternative sets of assumptions that the actuary judges to be reasonable.”
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Introduction (continued)
Goals of this paper
– Discuss concept of “Range of Reasonable Estimates”
– Describe methods for determining range
– Demonstrate a sound method for aggregation of line/year ranges
– Recommend a basis for application of range to individual decisions
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Range of Reasonable Estimates
“Reasonable” was unfortunate choice
– implies estimates outside range are “unreasonable”
– circularity in ASOP No. 36
– would have preferred:
- reasonable assumptions
- appropriate methodology
- actuarially sound estimates
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Range of Reasonable Estimates (continued)
Range arises from uncertainty associated with estimates
Range reflects both process and parameter variance
– Statement of Principles focuses on process variance
– ASOP No. 36 focuses on methods and assumptions
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Range of Reasonable Estimates (continued)
Range does not contain all possibilities
Range may not contain most likely result
– Example:
- .01 probability of $1 million IBNR
- .99 probability of $0 IBNR
- Expected IBNR = $10,000
- Actuary sets range at $10,000 to $50,000
- Range excludes mode ($0) and median ($0)
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Financial Condition and the Range of Reasonable Estimates
Materiality and potential impact influence what is “reasonable”
Return to our $0 or $1 million example
– Assume $1 billion surplus
- $0 reserve may be reasonable due to immateriality of $1 million loss
- $1 million reserve would be unreasonable
- Range (?) $0 - $20,000
– Assume $1 million surplus
- $0 reserve not reasonable
- $1 million reserve may be reasonable due to impact (insolvency)
- Range (?) $10,000 - $1,000,000
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Methods for Estimating Ranges
Assumed Allowable Deviations
Alternative Methods
Alternative Assumptions
Method of Convolutions
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Methods for Estimating Ranges – Assumed Allowable Deviations
Example ±5% of Total Needed Reserve (TNR)
– Assume TNR as follows:
- Lognormal
- mean = $1,000,000 (µ = 13.469)
- c.v. = 1.0 ( = .83255)
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Methods for Estimating Ranges – Assumed Allowable Deviations
Total Needed Reserve CDF
Best Estimate $1,000,000
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
$0 $1,000,000 $2,000,000 $3,000,000 $4,000,000 $5,000,000
Total Loss Reserves
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Methods for Estimating Ranges – Assumed Allowable Deviations
Range established as ±5% of Total Needed Reserve (TNR)
Low = $950,000, High = $1,000,000
Assumed Allowable Deviation 5% of TNR
$950,000 $1,050,000
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
$500,000 $1,000,000 $1,500,000
Total Loss Reserves
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Methods for Estimating Ranges – Assumed Allowable Deviations
Problems with method– Deviations should vary by line– Calculation of deviation equivalent to calculating range– Best estimate forced to midpoint
Assumed Allowable Deviation 5% of TNR
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
$0 $1,000,000 $2,000,000 $3,000,000 $4,000,000 $5,000,000
Total Loss Reserves
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Methods for Estimating Ranges – Alternative Methods
Most common method in practice today
Run multiple methods and use results to estimate range
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Methods for Estimating Ranges – Alternative Methods
$0
$200,000
$400,000
$600,000
$800,000
$1,000,000
$1,200,000Hindsight
FrequencySeverity
P aidDevelopment Best Estimate Range
IncurredDevelopment Cape Cod
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Methods for Estimating Ranges – Alternative Methods
Where methods are independent this is reasonable approach
Adding Bornhuetter-Ferguson to loss development and loss ratio methods provides no additional insight – only weight.
Line by line review essential to check for underlying changes (e.g. case reserve adequacy)
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Methods for Estimating Ranges – Alternative Assumptions
Actuary picks low (optimistic) and high (pessimistic) factors for each assumption
Results determine range
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Methods for Estimating Ranges – Alternative Assumptions
$0
$200,000
$400,000
$600,000
$800,000
$1,000,000
$1,200,000
$1,400,000
P aidOptimistic
IncurredOptimistic P aid Neutral
IncurredNeutral
BestEstimate Range
P aidP essimistic
IncurredP essimistic
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Methods for Estimating Ranges – Alternative Assumptions
This method tends to produce ranges which are too wide.
Individual age-to-age factors are not successively independent
Combination of many optimistic or pessimistic assumptions produces unreasonably low or high aggregations
There is a way to overcome problems…
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Methods for Estimating Ranges – Method of Convolutions
Consider a standard 5x5 development triangle
AccidentYear 12 24 36 48 601998 $1,503,839 $2,490,404 $4,266,948 $6,144,355 $6,266,5841999 1,535,773 3,028,897 4,874,340 7,348,5702000 1,989,915 3,574,304 5,790,8112001 1,660,687 3,031,9522002 2,224,336
Table 1
Case Incurred by Age
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Methods for Estimating Ranges – Method of Convolutions
Which gives rise to a 4x4 triangle of development factors
AccidentYear 12-24 24-36 36-48 48-601998 1.656 1.713 1.440 1.0201999 1.972 1.609 1.5082000 1.796 1.6202001 1.826
Table 2
Incremental Development Factors
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Methods for Estimating Ranges – Method of Convolutions
Assume all claims settled by age 60
Use “Chinese menu” method (“One from column A, …)– 4! (24) combinations for 2001 year– 3! (6) combinations for 2000 year– 2! (2) combinations for 1999 year– 1! (1) combination for 1998 year
24 x 6 x 2 x 1 = 288 combinations for aggregate ultimate loss
AccidentYear 12-24 24-36 36-48 48-601998 1.656 1.713 1.440 1.0201999 1.972 1.609 1.5082000 1.796 1.6202001 1.826
Table 2
Incremental Development Factors
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Methods for Estimating Ranges – Method of Convolutions
Produces Aggregate IBNR Distribution
Convoluted IBNR CDF
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
$13,000,000 $14,000,000 $15,000,000 $16,000,000 $17,000,000 $18,000,000
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Methods for Estimating Ranges – Method of Convolutions
Best Estimate (from average factors) between 53rd and 54th percentiles
Convoluted IBNR CDF
Best Estimate $15,303,099
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
$13,000,000 $14,000,000 $15,000,000 $16,000,000 $17,000,000 $18,000,000
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Methods for Estimating Ranges – Method of Convolutions
Example – select range from 10th to 90th percentile as reasonable
80% Range of Reasonable Estimates
Best Estimate $15,303,099
$16,360,336
$14,282,084
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
$13,000,000 $14,000,000 $15,000,000 $16,000,000 $17,000,000 $18,000,000
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Methods for Estimating Ranges – Method of Convolutions
In practice, several methods are convoluted
– Each method separately– Results combined into single distribution– Since different methods have different numbers of convolutions, must be
careful with weighting – e.g. loss ratio method
AY 2002 .680, .690, .700, .710, .720
AY 2001 .675, .680, .685, .690
AY 2000 .645, .650, .655
AY 1999 .678, .680
- 5 x 4 x 3 x 2 = 120 convolutions – must be doubled to roughly equal weight of 288 development factor convolutions
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Methods for Estimating Ranges – Method of Convolutions
Number of convolutions escalates quickly!
– individual values from a k x k development factor triangle (k+1 by k+1 loss triangle)
– 4 x 4 triangle: 288
– 8 x 8 triangle: 5,056,584,744,960,000
k
k1
!
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Methods for Estimating Ranges – Method of Convolutions
Limiting Number of Convolutions – One Method– Convolute “youngest” 4x4 triangle and use average for remainder– Example AY 7 as of age 3
- Convolute 3 to 7 development (4x4) and multiply by average 7-ult
Year 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-Ult1 Factor Factor Factor Factor2 Factor Factor Factor Factor CNV 1x13 Factor Factor Factor Factor4 Factor Factor Factor Factor5 Factor Factor Factor Factor6 Factor Factor Factor Avg. 8-Ult7 Factor Factor 8 Factor9
Avg. 7-UltAvg. 6-Ult.
CONVOLUTED 4x4CONVOLUTED 4x4
CONVOLUTED 3x3CONVOLUTED 2x2
Avg. 5-Ult
Age
CONVOLUTED 4x4CONVOLUTED 4x4
CONVOLUTED 4x4
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Methods for Estimating Ranges – Method of Convolutions
Limiting Number of Convolutions – One Method (continued)
– Method reduces convolutions for the 8x8 triangle to:
- 1! x 2! x 3! x 4! x 4! x 4! x 4! x 4! = 95,551,488
- Reasonable number for computer analysis
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Aggregation of Ranges
Recall that we are dealing with reasonable estimates, not possibilities
Lows, highs of component estimates cannot be added
Example: Four lines, four open accident years for each line
– Assume two reasonable estimates for each (“loway,l” and “highay,l”)
– Assume pr(loway,l) = pr(highay,l) = 50%
– Sum of reasonable lows is not a reasonable estimate
%001526.05.)low(pr 16,
l aylay
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Aggregation of Ranges
A Probability Approach
– Toss of ten true coins– Estimate number of “heads”– Reasonable range contains about 90%
Range = 3 to 7 heads– 89% probability
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Consider 10 groups of 10 coins
Aggregation of Ranges
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Reasonable (90%) range for number of heads in 100 coins– 42 to 58 heads (91% probability)
If we used the 3 to 7 range 10 times– 30 to 70 heads (99.997% probability)
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Aggregation of Ranges
A Proposed Method– Assume accident year selections are independent
– Assume line of business selections are independent
- Not strictly true, but reasonable when applied to most methods
– Assume width of range is k (where is standard deviation of estimates)
– Width of aggregate range is square root of sum of squares of individual widths
– Aggregate best estimate placement weighted average
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Aggregation of Ranges
Example
Acc Best RangeLine Year Low Est High Width Width2 Width2
A 1 $450 $500 $600 $150 $22,5002 $2,700 $3,000 $3,500 $800 $640,0003 $6,000 $7,000 $7,500 $1,500 $2,250,0004 $9,000 $11,000 $14,000 $5,000 $25,000,000
Total $21,500 $27,912,500 $5,283B 1 $90 $100 $115 $25 $625
2 $1,400 $1,500 $1,650 $250 $62,5003 $2,800 $3,000 $3,300 $500 $250,0004 $6,800 $7,500 $8,400 $1,600 $2,560,000
Total $12,100 $2,873,125 $1,695Total $33,600 $30,785,625 $5,548
Total Needed Reserve
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Aggregation of Ranges
Example (continued)
Best Position CalculatedAcc Est in Weighted Aggregate Aggregate Aggregate
Line Year Weight Range Position Position Low HighA 1 1.488% 0.3333 0.004960
2 8.929% 0.3750 0.0334823 20.833% 0.6667 0.1388894 32.738% 0.4000 0.130952
Total 63.988% 0.308284 0.481783 $18,955 $24,238B 1 0.298% 0.4000 0.001190
2 4.464% 0.4000 0.0178573 8.929% 0.4000 0.0357144 22.321% 0.4375 0.097656
Total 36.012% 0.152418 0.423244 $11,383 $13,078Total 100.000% 0.460702 0.460702 $31,044 $36,592
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Application of Ranges
ASOP No. 36
– “When the stated reserve amount is within the actuary’s range of reasonable estimates the actuary should issue a statement of actuarial opinion that the stated reserve amount makes a reasonable provision for the liabilities associated with the specified reserves.”
Statement of Principles
– Actuary should consider “both the relative likelihood of estimates within the range and the financial reporting context in which the reserve will be presented.”
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Application of Ranges
Where company has established the reserve independently of the opining actuary’s analysis (“untutored” reserve)
– ASOP No. 36 “stated reserve” language applies
Where company establishes reserve based upon opining actuary’s analysis
– Opining actuary now “owns” the estimate and the Statement of Principles language requires the reserves be at or above the opining actuary’s best estimate
Note that this is my opinion, not established doctrine.
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Conclusion
We must guard against the use of the concept of a “range of reasonable estimates” as justification for carrying reserves which we expect will be inadequate.