Transcript Ranges of Reasonable Estimates Charles L. McClenahan, FCAS, MAAA

Iowa Actuaries Club, February 9, 2004
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
– 1973 Robert Anker review described three ranges
- Absolute Range = Lowest indication to Highest indication
- Likely Range = Lowest selected to Highest Selected
- Best Estimate Range
2003 Casualty Loss Reserve Seminar
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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.”
Iowa Actuaries Club, February 9, 2004
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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.”
Iowa Actuaries Club, February 9, 2004
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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.”
Iowa Actuaries Club, February 9, 2004
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Introduction (continued)

Statement of Actuarial Opinion on P&C Loss Reserves
– Year-end 2004
- Actuary must include either a point estimate or a range of reasonable
estimates for gross and net reserves in the Actuarial Report
– Year-end 2005
- Confidential Actuarial Opinion Summary (AOS)
- Range (gross and net)
- Point-estimate (gross and net)
Iowa Actuaries Club, February 9, 2004
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Introduction (continued)

Goals of this presentation
– Discuss concept of “Range of Reasonable Estimates”
– Describe some methods for determining range
– Demonstrate a sound method for aggregation of line/year ranges
– Discuss some problems with application of ranges
– Recommend a basis for application of range to individual decisions
Iowa Actuaries Club, February 9, 2004
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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
Iowa Actuaries Club, February 9, 2004
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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)
Iowa Actuaries Club, February 9, 2004
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Some Methods for Estimating Ranges

Assumed Allowable Deviations

Alternative Methods

Alternative Assumptions
Iowa Actuaries Club, February 9, 2004
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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)
Iowa Actuaries Club, February 9, 2004
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Methods for Estimating Ranges –
Assumed Allowable Deviations
Total Needed Reserve CDF
100%
Cumulative Distribution
90%
80%
70%
60%
50%
Best Estimate
$1,000,000
40%
30%
20%
10%
0%
$0
$1,000,000
$2,000,000
$3,000,000
$4,000,000
$5,000,000
Total Loss Reserves
Iowa Actuaries Club, February 9, 2004
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Methods for Estimating Ranges –
Assumed Allowable Deviations

Range established as ±5% of Total Needed Reserve (TNR)
Low = $950,000, High = $1,050,000
Assumed Allowable Deviation 5% of TNR
100%
Cumulative Distribution
90%
80%
70%
60%
50%
40%
30%
$1,050,000
$950,000
20%
10%
0%
$500,000
$1,000,000
$1,500,000
Total Loss Reserves
Iowa Actuaries Club, February 9, 2004
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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
100%
Cumulative Distribution
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
$0
$1,000,000
$2,000,000
$3,000,000
$4,000,000
$5,000,000
Total Loss Reserves
Iowa Actuaries Club, February 9, 2004
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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
Iowa Actuaries Club, February 9, 2004
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Methods for Estimating Ranges –
Alternative Methods
Hindsight
Frequency
Severity
P aid
Develo pment B est Estimate
Range
Incurred
Develo pment
Cape Co d
$1,200,000
$1,000,000
$800,000
$600,000
$400,000
$200,000
$0
Iowa Actuaries Club, February 9, 2004
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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)
Iowa Actuaries Club, February 9, 2004
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Methods for Estimating Ranges –
Alternative Assumptions

Actuary picks low (optimistic) and high (pessimistic) factors for each
assumption

Results determine range
Iowa Actuaries Club, February 9, 2004
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Methods for Estimating Ranges –
Alternative Assumptions
P aid
Optimistic
Incurred
Optimistic
P aid Neutral
Incurred
Neutral
B est
Estimate
Range
P aid
Incurred
P essimistic P essimistic
$1,400,000
$1,200,000
$1,000,000
$800,000
$600,000
$400,000
$200,000
$0
Iowa Actuaries Club, February 9, 2004
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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…
Iowa Actuaries Club, February 9, 2004
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Methods for Estimating Ranges –
Method of Convolutions

Calculates all (or most) combinations of observed development factors

Application to latest diagonal gives rise to distribution of ultimate losses

First documented application circa 1992 by C. K. Stan Khury

Methodology discussed in my Fall 2003 CAS Forum paper
Iowa Actuaries Club, February 9, 2004
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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%
pr ( low ay,l )  .5  0.001526%
16
l
ay
– Sum of reasonable lows is not a reasonable estimate
Iowa Actuaries Club, February 9, 2004
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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
Iowa Actuaries Club, February 9, 2004
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Aggregation of Ranges

Consider 10 groups of 10 coins
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Aggregation of Ranges

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)
Iowa Actuaries Club, February 9, 2004
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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
Line
A
B
Total
Total Needed Reserve
Acc
Best
Year
Low
Est
1
$450
$500
2
$2,700
$3,000
3
$6,000
$7,000
4
$9,000 $11,000
Total
$21,500
1
$90
$100
2
$1,400
$1,500
3
$2,800
$3,000
4
$6,800
$7,500
Total
$12,100
$33,600
Iowa Actuaries Club, February 9, 2004
High
$600
$3,500
$7,500
$14,000
$115
$1,650
$3,300
$8,400
Range
Width2
Width
$150
$22,500
$800
$640,000
$1,500 $2,250,000
$5,000 $25,000,000
$27,912,500
$25
$625
$250
$62,500
$500
$250,000
$1,600 $2,560,000
$2,873,125
$30,785,625
Width2
$5,283
$1,695
$5,548
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Aggregation of Ranges

Example (continued)
Line
A
B
Acc
Year
1
2
3
4
Total
1
2
3
4
Total
Total
Iowa Actuaries Club, February 9, 2004
Best
Position
Est
in
Weight
Range
1.488%
0.3333
8.929%
0.3750
20.833%
0.6667
32.738%
0.4000
63.988%
0.298%
0.4000
4.464%
0.4000
8.929%
0.4000
22.321%
0.4375
36.012%
100.000%
Weighted
Position
0.004960
0.033482
0.138889
0.130952
0.308284
0.001190
0.017857
0.035714
0.097656
0.152418
0.460702
Calculated
Aggregate
Position
Aggregate Aggregate
Low
High
0.481783
$18,955
$24,238
0.423244
0.460702
$11,383
$31,044
$13,078
$36,592
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Problems With the Application of Ranges
Confusion about the concept
Some view “range of reasonable estimates” in the context of an individual
actuary –
Different assumptions, different methods, distribution of TNR
Iowa Actuaries Club, February 9, 2004
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Problems With the Application of Ranges
Individual Actuary - Reasonable Estimates
Range of Reasonable Estimates
$40,000
$45,000
Iowa Actuaries Club, February 9, 2004
$50,000
$55,000
$60,000
$65,000
$70,000
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Problems With the Application of Ranges
Confusion about the concept
Some view “range of reasonable estimates” in the context of the
universe of actuaries –
Different assumptions, different methods, distribution of “best estimates”
Iowa Actuaries Club, February 9, 2004
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Problems With the Application of Ranges
Multiple Actuaries - Best Estimates
"Best Estimate"
"Best Estimate"
Low Actuary
High Actuary
Range of Reasonable Estimates
$40,000
$45,000
Iowa Actuaries Club, February 9, 2004
$50,000
$55,000
$60,000
$65,000
$70,000
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Problems With the Application of Ranges
Consider the following hypothetical:
Appointed Actuary uses individual actuary interpretation
Range of Reasonable Estimates
Iowa Actuaries Club, February 9, 2004
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Problems With the Application of Ranges
Consider the following hypothetical:
Appointed Actuary uses individual actuary interpretation
Appointed Actuary is also “Low” Actuary
"Best Estimate"
Low Actuary
High Actuary
Iowa Actuaries Club, February 9, 2004
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Problems With the Application of Ranges
Minimum “clean opinion” reserve becomes low actuary’s low
Low Actuary as Appointed Actuary
"Best Estimate"
"Best Estimate"
Low Actuary
High Actuary
Range of Reasonable Estimates
$40,000
$45,000
Iowa Actuaries Club, February 9, 2004
$50,000
$55,000
$60,000
$65,000
$70,000
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Problems With the Application of Ranges
Many reserves judged reasonable by other actuaries outside range
Low Actuary as Appointed Actuary
"Best Estimate"
"Best Estimate"
Low Actuary
High Actuary
Range of Reasonable Estimates
"Unreasonable"
$40,000
$45,000
Iowa Actuaries Club, February 9, 2004
$50,000
$55,000
$60,000
$65,000
$70,000
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Application of Ranges in Individual Situations

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.”
Iowa Actuaries Club, February 9, 2004
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Application of Ranges in Individual Situations

A “Modest Proposal” (apologies to Jonathan Swift)
– Profession should agree that “range of reasonable estimates” be
interpreted as a range of alternative “best estimates” (multiple actuaries
approach)
– Concept of range should be used only in reviewing estimates of others, not
in establishing liabilities
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Application of Ranges in Individual Situations

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.
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For More Information

Fall 2003 CAS Forum
– Probabilistic Framework for Evaluating Materiality and Variability in Loss
Reserve Estimates – Bass & Khury
– Measurement of Reserve Variability - Hayne
– Estimation and Application of Ranges of Reasonable Estimates McClenahan
– Loss Reserve Estimates: A Statistical Approach for Determining
“Reasonableness” – Shapland
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