Roger M. Hayne, FCAS, MAAA
C.K. “Stan” Khury, FCAS, MAAA
Robert F. Wolf, FCAS, MAAA
2005 CAS Spring Meeting



Changes:
– Changes in the 2005 NAIC reporting requirements (best estimate, ranges, etc.)
– SEC pending rule changes about disclosures with respect to items involving uncertainty
– Pending changes in the reserving principles
– Pending changes in the ASOP
Unifying theme driving all of these changes:
– A reserve is really a probability statement consisting of an amount x plus the probability that the final settlement will not exceed x




Start simple – a range around what?
Accountants say it is to be a “reasonable estimate” of the unpaid claim costs
CAS says that “an actuarially sound loss reserve … is a provision, based on estimates derived from reasonable assumptions and appropriate methods…”





An “estimate” of amount unpaid
Is it an estimate of the average amount to be paid? No
Is it an estimate of the most likely amount to be paid? No
It is an estimate of the amount to be paid





Reserves as of 12/31/2005
Claim to be settled 1/1/2006 with immediate payment of $1 million times roll of fair die
All results equally likely so some accounting guidance says book low end ($1 million), others midpoint ($3.5 million)
Mean and median are $3.5 million






Reserves as of 12/31/2005
Claim to be settled 1/1/2006 as $1 million times toss of loaded die:
– Prob(x=1)=Prob(x=6)=1/4
– Prob(x=2)=Prob(x=5)=1/6
– Prob(x=3)=Prob(x=4)=1/12
What do you book now?
Mean and median still $3.5 million
“Most likely” is either $1 million or $6 million





Traditional actuarial methods:
– “Deestribution? We don’ need no steenkin’ deestribution.”
Traditional methods give “an estimate”
No assumptions, thus no conclusions on distributions
There are stochastic versions of some methods (chain ladder, Bornhuetter-
Ferguson)



Traditional methods give “estimates”
– Not estimates of the mean
– Not estimates of the median
– Not estimates of the mode
– Not estimates of a percentile
– Not estimates of any statistic of the distribution
– Just “estimates”
Distributions are normally possible only after added assumptions





Designed for traditional analysis
Does not address or even talk about distributions
Definition is “soft” and talks about results of “appropriate” methods under
“reasonable” assumptions
Does not refer to the distribution of potential outcomes




Range of reasonable results an attempt to quantify an actuary’s “gut feel” or
“judgment”
Typically you do a lot of methods
– If they “bunch up” you feel “good”
– If they are “spread out” you feel
“uncomfortable”
In the end – estimate is quite subjective




A method is a general approach
– Chain ladder
– Bornhuetter-Ferguson
A model usually specifies an underlying process or distribution and the focus is on identifying the parameters of the model
Most traditional actuarial forecasting approaches are methods and not models





Stochastic methods have assumptions about underlying models
Nearly all focus on a single data set (paid loss triangle, incurred loss triangle, etc.)
Do not directly model multiple sources of information (e.g. counts, paid, and incurred at the same time)
Mack/Quarg method not yet stochastic



Components of uncertainty:
– Process
– Parameter
– Model/Specification
Any true estimate of the distribution of outcomes ordinarily would recognize all three





Uncertainty that cannot be avoided
Inherent in the process
Example – the throw of a fair die
– You completely know the process
– You cannot predict the result with certainty
Usually the smallest component of insurance distributions (law of large numbers)





Uncertainty about the parameters of models
(Note: Some models are not parametric)
The underlying process is known
Just the position of some “knobs” is not
Example – flip of a weighted coin
– Uncertainty regarding the expected proportion of heads





The uncertainty that you have the right model to begin with
Not just what distributions, but what form the model should take
Most difficult to estimate
Arguably un-estimable for P&C insurance situations





Combines all sources of uncertainty
Gives potential future payments at point in time along with an associated likelihood
Must be estimated
Estimation is itself subject to uncertainty, so we are not away from
“reasonableness” issues






I use a series of methods
My “range of reasonable estimates” is the range of forecasts from the various methods
Is this reasonable?
What if one or more of the assumptions or methods is really “unreasonable”?
Is something outside this range necessarily
“unreasonable”?





Take largest and smallest forecast by accident year
Add these together
Is this a “reasonable range”
Example:
– Roll of single fair die, 2/3 confidence interval is between 2 an 5 inclusive
– Roll of a pair of fair dice, 2/3 confidence interval is between 5 and 9 inclusive, not 4 to 10 (5/6).






Your best estimate is $x
Actual future payments is $y (>$x)
Conclusion – you were “wrong”
Why? The myth that the estimate actually will happen
Problem – a reserve is a distribution, not just a single point, any other treatment is doomed to failure




Actually, why can’t the accountants get it right?
The accountants need to deal with the fact rather than the myth that the actual payments will equal the reserve estimate
Need to
– Be able to book a distribution
– Recognize the entire distribution
– Recognize context (company environment)
– Realize that future payments = reserves is an accident with a nearly 0% chance of happening




Why not set reserves so that the loss in company value when actual payments turn out different is the least expected
Note expectation taken over all possible reserve outcomes (along with their probabilities)
Economically rational – focuses on the impact of the final settlement on a company’s net worth



Since any single number will be “wrong” let me submit a reasonable estimate of reserves (compliments of Rodney Kreps)
Suppose
– (a really BIG suppose) we know the probability density function of future claim payments and expenses is f(x)
– For simplicity assume a one year time horizon
– g(x,μ) denotes the decrease in shareholder
(policyholder) value of the company if reserves are booked at μ but payments are actually x.


A rational reserve (i.e. “estimate of future payments”) is that value of μ that minimizes
P


  g
 x,
   
0

 i.e. the expected penalty for setting reserves at μ over all reserve outcomes






Likely not symmetric
Likely flat in a region “near” μ
Increases faster when x is above μ than when x is below
Likely increases at an increasing rate when x is above μ
Such a function generally gives an estimate above the mean

Distribution of IBNR Outcomes
Using Chain Ladder Method - UnWeighted Analysis
0.55%
0.50%
0.45%
0.40%
0.35%
0.30%
0.25%
0.20%
0.15%
0.10%
0.05%
0.00%
165 313 461 610 758 906
1,054 1,202 1,350 1,498 1,646 1,794 1,942 2,090 2,238 2,386 2,534 2,682 2,830 2,978 3,126 3,274 3,422 3,570 3,718 3,866 4,014 4,162 4,310 4,459 4,607 4,755
IBNR Amounts
BassAndKhury.com

Distribution of IBNR Outcomes
Using Chain Ladder Method - Incurred - UnWeighted Analysis
0.65%
0.60%
0.55%
0.50%
0.45%
0.40%
0.35%
0.30%
0.25%
0.20%
0.15%
0.10%
0.05%
0.00%
-3.1 -1.7 -0.3 1.2
2.6
4.0
5.5
6.9
8.4
9.8 11.2 12.7 14.1 15.6 17.0 18.4 19.9 21.3 22.7 24.2 25.6 27.1 28.5 29.9 31.4
IBNR Amounts
BassAndKhury.com