From “Reasonable Reserve Range” to “Carried Reserve” – What do you Book? 2007 CAS Annual Meeting Chicago, Illinois November 11-14, 2007 Mark R. Shapland, FCAS, ASA, MAAA Consulting Actuary

2  Terminology  Ranges vs. Distributions  What is “Reasonable” to Book? Overview

3  Reserve – an amount carried in the liability section of a risk-bearing entity’s balance sheet for claims incurred prior to a given accounting date.  Liability – the actual amount that is owed and will ultimately be paid by a risk-bearing entity for claims incurred prior to a given accounting date.  Loss Liability – the expected value of all estimated future claim payments.  Risk (from the “risk-bearers” point of view) – the uncertainty (deviations from expected) in both timing and amount of the future claim payment stream. Terminology

4  Process Risk – the randomness of future outcomes given a known distribution of possible outcomes.  Parameter Risk – the potential error in the estimated parameters used to describe the distribution of possible outcomes, assuming the process generating the outcomes is known.  Model Risk – the chance that the model (“process”) used to estimate the distribution of possible outcomes is incorrect or incomplete. Terminology

5  Risk – unknown outcomes with quantifiable probabilities.  Uncertainty – unknown outcomes that cannot be estimated or quantified.  All entrepreneurship involves both risk (which can be transferred) and uncertainty (which cannot be transferred). Terminology Source: Knight, Frank H “Risk, Uncertainty, and Profit.” Houghton Mifflin.

6  Variance, standard deviation, kurtosis, average absolute deviation, Value at Risk, Tail Value at Risk, etc. which are measures of dispersion.  Other measures useful in determining “reasonableness” could include: mean, mode, median, pain function, etc.  The choice for measure of risk will also be important when considering the “reasonableness” and “materiality” of the reserves in relation to the capital position. Terminology  Measures of Risk from Statistics:

7  As part of your analysis of unpaid claims, the CEO of your company would like you to create a “Reserve Range” to discuss with the Board.  How do you proceed? CAS Exams vs. Real World

8 $11M $16M  You use a variety of methods to arrive at a range of point estimates.  Removing the “unreasonable” estimates you are left with the following:

9  You present the “Reserve Range” to the CEO and the Board and they ask:  How likely are these reserves to be adequate?  How likely is it that the final result will be in this range?  How much capital do we need to support our business?  OOPS!!! CAS Exams vs. Real World

10  A “Range” is not the same as a “Distribution”  A 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.  A Distribution is a statistical function that attempts to quantify probabilities of all possible outcomes. Ranges vs. Distributions

11 Ranges vs. Distributions  A Range, by itself, creates problems:  A range can be misleading to the layperson – it can give the impression that any number in that range is equally likely.  A range can give the impression that as long as the carried reserve is “within the range” anything is reasonable.

12 Ranges vs. Distributions  A Range, by itself, creates problems:  There is currently no specific guidance on how to consistently determine a range within the actuarial community (e.g., +/- X%, +/- $X, using various estimates, etc.).  A range, in and of itself, needs some other context to help define it (e.g., how to you calculate a risk margin?)

13 Ranges vs. Distributions  A Distribution provides:  Information about “all” possible outcomes.  Context for defining a variety of other measures (e.g., risk margin, materiality, risk based capital, etc.)

14  A Method is an algorithm or recipe – a series of steps that are followed to give an estimate of future payments.  The well known chain ladder (CL) and Bornhuetter-Ferguson (BF) methods are examples.  The search for the “best” pattern. Methods vs. Models

15  A Model specifies statistical assumptions about the loss process, usually leaving some parameters to be estimated.  Then estimating the parameters gives an estimate of the ultimate losses and some statistical properties of that estimate.  The search for the “best” model. Methods vs. Models

16 What Reserve to Book? $11M $16M (Point Estimates)

17 The Most Likely Outcome? What Reserve to Book? Mode (Distributions)

18 Equally Likely to be High / Low? What Reserve to Book? Mode Median (Distributions)

19 Enough On Average? What Reserve to Book? Mode Median Expected Value (Distributions)

20 Which Risks are Included? What Reserve to Book? Mode Median Expected Value (Distributions)

21 What Reserve to Book? LOB “A” LOB “B” LOB “C” Aggregate Distribution with 100% Correlation (Added) Aggregate Distribution with 0% Correlation (Independent) Aggregate Distributions

22 What Reserve to Book? Aggregate Distribution with 100% Correlation (Added) Aggregate Distribution with 0% Correlation (Independent) Expected Value 99 th Percentile Capital = 1,000M Capital = 600M Aggregate Distributions

23 What Reserve to Book? $11M $16M

24 Mode Median Expected Value 50 th Percentile 75 th Percentile What Reserve to Book? $11M $16M

25 Mode Median Expected Value 50 th Percentile 75 th Percentile What Reserve to Book? $11M $16M

26 Mode Median Expected Value 50 th Percentile 75 th Percentile What Reserve to Book? $11M $16M

27 Mode Median Expected Value 50 th Percentile 75 th Percentile What Reserve to Book? $11M $16M

28 What Reserve to Book? Range of Reasonable Estimates Reasonable Distribution

29 What Reserve to Book? $11M $16M (Range of “Reasonable” Estimates) “Best” Estimate Range of Possible Estimates

30 Liability Estimates Probability What Reserve to Book? (Multiple Distributions)

31 What Reserve to Book? Range of Mean Estimates (Reasonable Distributions) Unpaid Estimates Probability

32 What Reserve to Book? Range of Mean Estimates “Best” Distribution (Reasonable Distributions) Unpaid Estimates Probability

33 Questions?

34 CAS Exams – Revisited  Reserves as of 12/31/2006?  One claim to be settled 1/1/2007 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), still others the mode ($1, $2, $3, $4, $5 or $6 million)  Mean and median are $3.5 million  What reserve to book?