Reserve Uncertainty

1
Reserve Uncertainty

• byRoger M. Hayne, FCAS, MAAAMilliman USA CAS
  Meeting
• May 18-21, 2003

2
Reserves Are Uncertain?

• Reserves are just numbers in a financial
  statement
• What do we mean by reserves are uncertain?
• Numbers are estimates of future payments
• Not estimates of the average
• Not estimates of the mode
• Not estimates of the median
• Not really much guidance in guidelines
• Rodney Kreps has more to say on this subject

3
Lets Move Off the Philosophy

• Should be more guidance in accounting/actuarial
  literature
• Not clear what number should be booked
• Less clear if we do not know the distribution of
  that number
• There may be an argument that the more uncertain
  the estimate the greater the margin
• Need to know distribution first

4
Traditional Methods

• Many traditional reserve methods are somewhat
  ad-hoc
• Oldest, probably development factor
• Fairly easy to explain
• Subject of much literature
• Not originally grounded in theory, though some
  have tried recently
• Known to be quite volatile for less mature
  exposure periods

5
Traditional Methods

• Bornhuetter-Ferguson
• Overcomes volatility of development factor method
  for immature periods
• Needs both development and estimate of the final
  answer (expected losses)
• No statistical foundation
• Frequency/Severity (Berquist, Sherman)
• Also ad-hoc
• Volatility in selection of trends averages

6
Traditional Methods

• Not usually grounded in statistical theory
• Fundamental assumptions not always clearly stated
• Often not amenable to directly estimate
  variability
• Traditional approach usually uses various
  methods, with different underlying assumptions,
  to give the actuary a sense of variability

7
Basic Assumption

• When talking about reserve variability primary
  assumption is
• Given current knowledge there is a
  distribution of possible future payments
  (possible reserve numbers)
• Keep this in mind whenever answering the question
  How uncertain are reserves?

8
Some Concepts

• Baby steps first, estimate a distribution
• Sources of uncertainty
• Process (purely random)
• Parameter (distributions are correct but
  parameters unknown)
• Specification/Model (distribution or model not
  exactly correct)
• Keep in mind whenever looking at methods that
  purport to quantify reserve uncertainty

9
Why Is This Important?

• Consider usual development factor projection
  method, Cik accident year i, paid by age k
• Assume
• There are development factors fi such that
• E(Ci,k1Ci1, Ci2,, Cik) fk Cik
• Ci1, Ci2,, CiI, Cj1, Cj2,, CjI independent
  for i ? j
• There are constants ?k such that
• Var(Ci,k1Ci1, Ci2,, Cik) Cik ?k2

10
Conclusions

• Following Mack (ASTIN Bulletin, v. 23, No. 2, pp.
  213-225)

• are unbiased estimates for the development
  factors fi
• Can also estimate standard error of reserve

11
Conclusions

• Estimate of mean squared error (mse) of reserve
  forecast for one accident year

12
Conclusions

• Estimate of mean squared error (mse) of the total
  reserve forecast

13
Sounds Good -- Huh?

• Relatively straightforward
• Easy to implement
• Gets distributions of future payments
• Job done -- yes?
• Not quite
• Why not?

14
An Example

• Apply method to paid and incurred development
  separately
• Consider resulting estimates and errors
• What does this say about the distribution of
  reserves?
• Which is correct?

15
Real Life Example

• Paid and Incurred as in handouts (too large for
  slide)
• Results

Paid Incurred
Case Reserve 96,917
Reserve Est. 358,453 90,580
s.e.(Est.) 41,639 13,524
16
A Real Life Example
17
A Real Life Example
18
What Happened?

• Conclusions follow unavoidably from assumptions
• Conclusions contradictory
• Thus assumptions must be wrong
• Independence of factors? Not really (there are
  ways to include that in the method)
• What else?

19
What Happened?

• Obviously the two data sets are telling different
  stories
• What is the range of the reserves?
• Paid method?
• Incurred method?
• Extreme from both?
• Something else?
• Main problem -- the method addresses only one
  method under specific assumptions

20
What Happened?

• Not process (that is measured by the
  distributions themselves)
• Is this because of parameter uncertainty?
• No, can test this statistically (from normal
  distribution theory)
• If not parameter, what? What else?
• Model/specification uncertainty

21
Why Talk About This?

• Most papers in reserve distributions consider
• Only one method
• Applied to one data set
• Only conclusion distribution of results from a
  single method
• Not distribution of reserves

22
Discussion

• Some proponents of some statistically-based
  methods argue analysis of residuals the answer
• Still does not address fundamental issue model
  and specification uncertainty
• At this point there does not appear much (if
  anything) in the literature with methods
  addressing multiple data sets

23
Moral of Story

• Before using a method, understand underlying
  assumptions
• Make sure what it measures what you want it to
• The definitive work may not have been written yet
• Casualty liabilities very complex, not readily
  amenable to simple models

24
All May Not Be Lost

• Not presenting the definitive answer
• More an approach that may be fruitful
• Approach does not necessarily have single model
  problems in others described so far
• Keeps some flavor of traditional approaches
• Some theory already developed by the CAS
  (Committee on Theory of Risk, Phil Heckman,
  Chairman)

25
Collective Risk Model

• Basic collective risk model
• Randomly select N, number of claims from claim
  count distribution (often Poisson, but not
  necessary)
• Randomly select N individual claims, X1, X2, ,
  XN
• Calculate total loss as T ? Xi
• Only necessary to estimate distributions for
  number and size of claims
• Can get closed form expressions for moments
  (under suitable assumptions)

26
Adding Parameter Uncertainty

• Heckman Meyers added parameter uncertainty to
  both count and severity distributions
• Modified algorithm for counts
• Select ? from a Gamma distribution with mean 1
  and variance c (contagion parameter)
• Select claim counts N from a Poisson distribution
  with mean ? ?
• If c lt 0, N is binomial, if c gt 0, N is negative
  binomial

27
Adding Parameter Uncertainty

• Heckman Meyers also incorporated a global
  uncertainty parameter
• Modified traditional collective risk model
• Select ? from a distribution with mean 1 and
  variance b
• Select N and X1, X2, , XN as before
• Calculate total as T ? ? Xi
• Note ? affects all claims uniformly

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Why Does This Matter?

• Under suitable assumptions the Heckman Meyers
  algorithm gives the following
• E(T) E(N)E(X)
• Var(T) ?(1b)E(X2)?2(bcbc)E2(X)
• Notice if bc0 then
• Var(T) ?E(X2)
• Average, T/N will have a decreasing variance as
  E(N)? is large (law of large numbers)

29
Why Does This Matter?

• If b ? 0 or c ? 0 the second term remains
• Variance of average tends to (bcbc)E2(X)
• Not zero
• Otherwise said No matter how much data you have
  you still have uncertainty about the mean
• Key to alternative approach -- Use of b and c
  parameters to build in uncertainty

30
If It Were That Easy

• Still need to estimate the distributions
• Even if we have distributions, still need to
  estimate parameters (like estimating reserves)
• Typically estimate parameters for each exposure
  period
• Problem with potential dependence among years
  when combining for final reserves

31
An Example

• Consider the data set included in the handouts
• This is hypothetical data but based on a real
  situation
• It is residual bodily injury liability under
  no-fault
• Rather homogeneous insured population

32
An Example(Continued)

• Applied several traditional actuarial methods
• Usual development factor
• Berquist/Sherman
• Hindsight reserve method
• Adjustments for
• Relative case reserve adequacy
• Changes in closing patterns

33
An Example(Continued)
Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method
Accident Paid Paid Paid Paid Adjusted CD Adjusted Paid CD Adjusted Paid CD Adjusted Paid CD Adjusted Paid
Year Incurred Devel. Sev. Pure Prem. Hindsight Incurred Devel. Sev. Pure Prem. Hindsight
1986 744 2,143 1,760 1,909 1,687 394 1,936 1,842 1,950 675
1987 2,335 6,847 5,583 5,128 5,128 2,348 6,000 5,790 5,220 2,301
1988 8,371 19,768 16,246 13,451 14,428 10,391 17,352 16,433 13,399 8,001
1989 25,787 44,631 36,887 29,232 32,199 26,048 39,241 36,431 28,512 19,174
1990 60,211 83,760 73,987 61,846 62,974 55,734 79,667 70,246 57,192 43,286
1991 83,093 130,907 95,283 95,185 78,616 79,573 154,268 87,625 84,688 72,157
34
An Example(Continued)

• Now review underlying claim information
• Make selections regarding the distribution of
  size of open claims for each accident year
• Based on actual claim size distributions
• Ratemaking
• Other
• Use this to estimate contagion (c) value

35
An Example(Continued)
Accident Reserve Reserve Unpaid Single Claim Single Claim Implied
Year Selected Std. Dev. Counts Average Std. Dev. c Value
1986 1,357 637 106 12,802 18,913 0.190
1987 4,260 1,620 330 12,909 19,072 0.135
1988 12,866 3,525 926 13,894 20,527 0.072
1989 30,212 6,428 1,894 15,951 23,566 0.044
1990 62,516 10,198 3,347 18,678 27,595 0.026
1991 90,014 19,166 4,071 22,111 32,666 0.045
36
An Example(Continued)

• Thus variation among various forecasts helps
  identify parameter uncertainty for a year
• Still global uncertainty that affects all years
• Measure this by noise in underlying severity

37
An Example(Continued)
Accident Severity Severity Estimate
Year Selected Fitted of 1/b
1986 7,723 7,780 0.993
1987 8,501 8,196 1.037
1988 9,577 8,634 1.109
1989 9,919 9,095 1.091
1990 10,739 9,581 1.121
1991 12,194 10,093 1.208
Variance 0.019
38
An Example(Continued)
39
CAS To The Rescue

• Still assumed independence
• CAS Committee on Theory of Risk commissioned
  research into
• Aggregate distributions without independence
  assumptions
• Aging of distributions over life of an exposure
  year
• Will help in reserve variability
• Sorry, do not have all the answers yet