by
Roger M. Hayne, FCAS, MAAA
Milliman USA
CAS Meeting
May 18-21, 2003
Milliman USA




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
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Let’s 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
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 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
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 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
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 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
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 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?”
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 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
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 Consider “usual” development factor projection method, paid by age k
C ik accident year i,
 Assume:
–
–
–
There are development factors f i
E( C i,k+1
| C i1
, C i2
,…, C ik
)= f k such that
C ik
{ C i1
, C i2
,…, C iI
}, { C j1
, C j2
,…, C jI
} independent for i  j
There are constants
 k such that
Var( C i,k+1
| C i1
, C i2
,…, C ik
)= C ik
 k
2
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 Following Mack ( ASTIN Bulletin, v. 23, No.
2, pp. 213-225) f
ˆ k
 j


1
C j k

1


1 j
C are unbiased estimates for the development factors f i
 Can also estimate standard error of reserve
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 Estimate of mean squared error ( mse ) of reserve forecast for one accident year:
  i

C
ˆ
2 iI

1 I  k I i
 f
ˆ
ˆ k k
2
2




C
1 ik

1 j


1
C jk




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 Estimate of mean squared error ( mse ) of the total reserve forecast:
 
 i
I 

2




 s.e.
  i
2


C
ˆ iI


I  j i 1
C
ˆ jI


I

1  k I i
2

I
ˆ

1 k
2 n


1
C nk f
ˆ k
2




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 Relatively straightforward
 Easy to implement
 Gets distributions of future payments
 Job done -- yes?
 Not quite
 Why not?
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 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?
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 Paid and Incurred as in handouts
(too large for slide)
 Results
Paid
Case Reserve
Reserve Est.
$358,453 s.e.(Est.) 41,639
Incurred
$96,917
90,580
13,524
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100,000 150,000 200,000 250,000 300,000 350,000 400,000 450,000 500,000
Paid Incurred
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100,000 150,000 200,000 250,000 300,000 350,000 400,000 450,000 500,000
Paid Incurred Actual
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 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?
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 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
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 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
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 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
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 Some proponents of some statisticallybased 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
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 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
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 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)
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

Basic collective risk model:
–
–
–
Randomly select N, number of claims from claim count distribution (often Poisson, but not necessary)
Randomly select N individual claims, X
1
, X
2
, …,
X
N
Calculate total loss as T =

X i
Only necessary to estimate distributions for number and size of claims
 Can get closed form expressions for moments (under suitable assumptions)
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 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 < 0, N is binomial, if c > 0, N is negative binomial
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 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 X
1
, X
2
, …, X
Calculate total as T =

N

X i as before
Note
 affects all claims uniformly
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 Under suitable assumptions the
Heckman & Meyers algorithm gives the following:
–
–
E( T ) = E( N )E( X )
Var( T )=

(1 +b )E( X 2 )+

2 ( b + c + bc )E 2 ( X )
 Notice if b = c =0 then
– Var( T )=

E( X 2 )
– Average, T / N will have a decreasing variance as E( N )=
 is large (law of large numbers)
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 If b

0 or c

0 the second term remains
 Variance of average tends to
( b + c + bc )E 2 ( 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
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 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
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 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
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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
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An Example
(Continued)
Accident
Year Incurred
1986
1987
1988
1989
1990
1991
744
2,335
8,371
Devel.
2,143
Sev.
1,760
6,847 5,583
19,768 16,246
25,787
60,211
44,631
83,760
36,887
73,987
83,093 130,907 95,283
Reserve Estimates by Method
Paid
Pure Prem.
Hindsight
Adjusted
Incurred Devel.
1,909
5,128
13,451
1,687
5,128
14,428
394
2,348
10,391
1,936
CD Adjusted Paid
Sev.
1,842
6,000 5,790
17,352 16,433
Pure Prem.
1,950
5,220
13,399
Hindsight
675
2,301
8,001
29,232
61,846
95,185
32,199
62,974
78,616
26,048
55,734
39,241
79,667
36,431
70,246
79,573 154,268 87,625
28,512
57,192
84,688
19,174
43,286
72,157
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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
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An Example
(Continued)
Accident Reserve Unpaid Single Claim Implied
Year Selected Std. Dev.
Counts Average Std. Dev.
c Value
1986
1987
1,357
4,260
1988 12,866
637
1,620
3,525
106
330
926
12,802
12,909
13,894
18,913
19,072
20,527
0.190
0.135
0.072
1989 30,212
1990 62,516
1991 90,014
6,428
10,198
19,166
1,894
3,347
4,071
15,951
18,678
22,111
23,566
27,595
32,666
0.044
0.026
0.045
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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
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An Example
(Continued)
Accident Severity
Year Selected
1986 7,723
1987 8,501
1988
1989
1990
1991
Variance
Fitted
7,780
8,196
9,577
9,919
8,634
9,095
10,739 9,581
12,194 10,093
Estimate of 1/

0.993
1.037
1.109
1.091
1.121
1.208
0.019
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An Example
(Continued)
100,000 150,000 200,000 250,000
With Uncertainty Without Uncertainty Actual
300,000
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 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
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