Estimation and Application of Ranges of Reasonable Estimates Charles L. McClenahan, FCAS, MAAA

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



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.”



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.”



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.”



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


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


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


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)


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


Methods for Estimating Ranges

Assumed Allowable Deviations

Alternative Methods

Alternative Assumptions

Method of Convolutions


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)



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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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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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



$0

$200,000

$400,000

$600,000

$800,000

$1,000,000

$1,200,000Hindsight

FrequencySeverity

P aidDevelopment Best Estimate Range

IncurredDevelopment Cape Cod



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)


Methods for Estimating Ranges – Alternative Assumptions

Actuary picks low (optimistic) and high (pessimistic) factors for each assumption

Results determine range



$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



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…


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



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



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



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



Best Estimate (from average factors) between 53rd and 54th percentiles

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



Example – select range from 10th to 90th percentile as reasonable

80% Range of Reasonable Estimates


$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



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



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

!



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


Avg. 5-Ult

Age


CONVOLUTED 4x4



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


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



A Probability Approach

– Toss of ten true coins– Estimate number of “heads”– Reasonable range contains about 90%

Range = 3 to 7 heads– 89% probability


Consider 10 groups of 10 coins



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)



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



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



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


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.”


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.


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.

Documents

Estimation and Application of Ranges of Reasonable Estimates Charles L. McClenahan, FCAS, MAAA