1. Session II: Reserve Ranges Who Does What
9/17/2018
Session II: Reserve Ranges Who Does What
Presented by
Roger M. Hayne, FCAS, MAAA
May 8, 2006

2. Reserve – A Loaded Term “Reserve” is an accounting term
A “reserve” to be a “reasonable estimate” of the unpaid claim costs
Definition is vague at best
Only requires a “reasonable estimate” (whatever that is)
Seems to imply an amount that will happen
Even if we knew entire process this is not much help in what to record
One number is not enough – the “reserve” is really a set of outcomes with related probabilities, a distribution
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3. Why A Distribution? Future payments are uncertain
A single number cannot convey that uncertainty
If we had a distribution then concepts like mean, mode, median, probability level, value at risk, or any other statistic have specific meaning
If accountants insist on a single number embodying all that “reserve” means then we can talk about which of (infinitely) many numbers best convey that message
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4. Our Solution Our solution – punt
We will try not to talk about “reserves”
We will try to focus on the “distribution of outcomes” under a policy or group of policies (or for an insurer)
The “distribution of outcomes” will inform the reserve to be recorded
Better having the “what to book” discussion with this knowledge than without it
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5. How Did We Get Here?
Accounting definition seems to be intentionally vague
Current CAS Statement of Principles on reserves also somewhat vague
Reasonable estimates
Ranges of reasonable estimates
No mathematical/statistical precision
To quote Tevye – “Tradition”
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6. Tradition Traditional actuarial forecast methods are deterministic
Do not have an underlying model (more on this later)
They produce “estimates”
Not estimates of the expected (mean)
Not estimates of the most likely (mode)
Not estimates of the middle-of-the-road (median)
Just “estimates” without quantifying variability or uncertainty
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7. Measuring Uncertainty In Times Past
Traditional methods do not give underlying distributions
Our “Fore-parents” knew this and tradition included the application of a variety of methods
Bunching up of methods gave a sense of variability or uncertainty
Methods gave similar answers => little uncertainty
Methods gave disperse answers => much uncertainty
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8. Reasonable Ranges and Ranges of Reasonable Results
Need for “Range of Reasonable Estimates”
Still important to discuss uncertainty
Still a need to quantify how uncertain an estimate is
Lack of statistical qualities in traditional forecasts
Solution “Range of Reasonable Estimates”
Definition still “soft” without any statistical meaning
Depends on nebulous term “reasonable”
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9. A Hint of the Future? Consider a more reasoned approach
Assume that a “reserve” still needs to be a single number but that (big assumption here) we all agree on which statistic to use
(Presenter’s digression – I like Rodney Kreps’ “Least Pain” statistic)
What statistical sense do terms like “reasonable estimate” and “range of reasonable estimates” convey?
To help let’s define a few terms
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10. Talking About Uncertainty
Ultimate future payments on insurance claims are generally unknown
Theoretically, for a given amount there is a probability that future payments will not exceed that amount
Problem, we usually need to estimate those probabilities
The way we do this can (should?) involve several steps
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11. Simple Example Write policy 1/1/2006, roll fair die and hide result
Reserves as of 12/31/2006
Claim to be settled 1/1/2007 with immediate payment of $1 million times the number already rolled
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, there is no mode
What would you book as a reserve?
Note here there is no model or parameter uncertainty
If only one statistic is “reasonable” then “range of reasonable estimates” is a single point
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12. Almost Simple Example Prob(x=1)=Prob(x=6)=1/4 Prob(x=2)=Prob(x=5)=1/6
Claim process as before
This time die is not fair:
Prob(x=1)=Prob(x=6)=1/4
Prob(x=2)=Prob(x=5)=1/6
Prob(x=3)=Prob(x=4)=1/12
Mean and median still $3.5 million
“Most likely” is either $1 million or $6 million
What do you book now?
The means are the same but is the reality?
Still no parameter or model uncertainty
Again, if only one statistic is “reasonable” then “range of reasonable estimates” is a single point
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13. Steps In Estimating
Define one or more models of the future payment process
Estimate the parameters underlying the model(s)
Assess the volatility of the process under the assumption that the model(s) and parameters are all correct
Aggregate the uncertainty from each of these steps
Particular contributions called respectively model/specification, parameter and process uncertainty
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14. Some Context
The aggregate distribution you get in the end is useful in talking about the “range of potential outcomes”
The “range of reasonable results” is not this range of potential outcomes
If one defines a particular statistic (mean, “least pain,” value at risk, etc.) as a “reasonable” reserve estimate then it makes sense to look at the distribution of that statistic under different selections of models and parameters
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15. Simple Inclusion of Parameter Uncertainty
Adding parameter uncertainty is not that difficult
Very simple example
Losses have lognormal distribution, parameters m (unknown) and σ2 (known), respectively the mean and variance of the related normal
The parameter m itself has a normal distribution with mean μ and variance τ2
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16. Simple Example Continued
Expected (“reasonable estimate”) is lognormal
Parameters μ+ σ2/2 and τ2
c.v.2 of expected is exp(τ2)-1
“What will happen” (“possible outcome”) is lognormal
Parameters μ and σ2+τ2
c.v.2 is exp(σ2+τ2)-1
c.v. = standard deviation/mean, measure of relative dispersion
Note expected is much more certain (smaller c.v.) than “what will happen”
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17. Carry the Same Thought Further
Suppose that judgmentally or otherwise we can quantify the likelihood of various models
Think of each of them as different possible future states of the world
Why not use this information similar to the way the normal distribution was used in the example to quantify parameter/model uncertainty
Simplifies matters
Quantifies relative weights
Provides for a way to evolve those weights
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18. An Evolutionary (Bayesian) Model
Again take a very simple example
Use the die example
For simplicity assume we book the mean
This time there are three different dice that can be thrown and we do not know which one it is
Currently no information favors one die over others
The dice have the following chances of outcomes: 1 2 3 4 5 6
1/6
1/21
2/21
3/21
4/21
5/21
6/21
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19. Evolutionary Approach
“What will happen” is the same as the first die, equal chances of 1 through 6
The expected has equally likely chances of being 2.67, 3.50, or 4.33
If you set your reserve at the “average” both have the same average, 3.5, the true average is within 0.83 of this amount with 100% confidence
There is a 1/3 chance the outcome will be 2.5 away from this pick.
We now “observe” a 2 – what do we do?
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20. Posterior(model|data)likelihood(data|model)prior(model)
How Likely Is It?
Likelihood of observing a 2:
Distribution 1 1/6
Distribution 2 2/21
Distribution 3 5/21
Given our distributions it seems more likely that the true state of the world is 3 (having observed a 2) than the others
Use Bayes Theorem to estimate posterior likelihoods
Posterior(model|data)likelihood(data|model)prior(model)
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21. Evolutionary Approach
Revised prior is now:
Distribution
Distribution
Distribution
Revised posterior distribution is now: 1 2 3 4 5 6
0.20
0.19
0.17
0.16
0.15
0.13
Overall mean is 3.3
The expected still takes on the values 2.67, 3.50, and 4.33 but with probabilities 0.48, 0.33, and 0.19 respectively (our “range”)
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22. Next Iteration Second observation of 1
Revised prior is now (based on observing a 2 and a 1):
Distribution
Distribution
Distribution
Revised posterior distribution is now: 1 2 3 4 5 6
0.25
0.21
0.18
0.15
0.12
0.09
Now the mean is 3.0
The expected can be 2.67, 3.50, or 4.33 with probability 0.67, 0.28, and 0.05 respectively
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23. Not-So-Conclusive Example
Observations 3, 4, 3, 4
Revised prior is now :
Distribution
Distribution
Distribution
Revised posterior distribution is unchanged from the start: 1 2 3 4 5 6
1/6
As are the overall mean and chances for various states
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24. Summary
Though our publics seem to want certainty future payments are uncertain
It is virtually certain actual future payments will differ from any estimate
Quantifying the distribution of future payments will inform discussion
Keep process, parameter and model/specification uncertainty in mind
Models contain more information than methods
“Ranges of reasonable estimates” are different than “ranges of possible outcomes”
9/17/2018