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Reserve Uncertainty

byRoger M. Hayne, FCAS, MAAAMilliman USACasualty Loss Reserve SeminarSeptember 10-11, 2001

Milliman USA

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

Milliman USA

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

Milliman USA

“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

Milliman USA

“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

Milliman USA

“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

Milliman USA

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?”

Milliman USA

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

Milliman USA

Why Is This Important?

- Consider “usual” development factor projection method, Cikaccident year i, paid by age k
- Assume:
- There are development factors fisuch that

E(Ci,k+1|Ci1, Ci2,…, Cik)= fkCik

- {Ci1, Ci2,…, CiI}, {Cj1, Cj2,…, CjI} independent for i j
- There are constants ksuch that

Var(Ci,k+1|Ci1, Ci2,…, Cik)= Cik k2

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- 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

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Conclusions

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

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Sounds Good -- Huh?

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

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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?

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A “Real Life” Example

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A “Real Life” Example

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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?

Milliman USA

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

Milliman USA

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

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Why Talk About This?

- Almost every paper in reserve distributions considers
- 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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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

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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

Milliman USA

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)

Milliman USA

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)

Milliman USA

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 < 0, N is binomial, if c > 0, N is negative binomial

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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)= (1+b)E(X2)+2(b+c+bc)E2(X)
- Notice if b=c=0 then
- Var(T)= E(X2)
- Average, T/N will have a decreasing variance as E(N)= is large (law of large numbers)

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Why Does This Matter?

- If b 0 or c 0 the second term remains
- Variance of average tends to (b+c+bc)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

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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

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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

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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)

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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)

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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)

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An Example(Continued)

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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

Milliman USA

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