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Reserve Uncertainty

- byRoger M. Hayne, FCAS, MAAAMilliman USA CAS

Meeting - May 18-21, 2003

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

Lets 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

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

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

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

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?

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

Why Is This Important?

- Consider usual development factor projection

method, Cik accident year i, paid by age k - Assume
- There are development factors fi such that
- E(Ci,k1Ci1, Ci2,, Cik) fk Cik
- Ci1, Ci2,, CiI, Cj1, Cj2,, CjI independent

for i ? j - There are constants ?k such that
- Var(Ci,k1Ci1, Ci2,, Cik) Cik ?k2

Conclusions

- 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

Conclusions

- Estimate of mean squared error (mse) of reserve

forecast for one accident year

Conclusions

- Estimate of mean squared error (mse) of the total

reserve forecast

Sounds Good -- Huh?

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

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?

Real Life Example

- Paid and Incurred as in handouts (too large for

slide) - Results

Paid Incurred

Case Reserve 96,917

Reserve Est. 358,453 90,580

s.e.(Est.) 41,639 13,524

A Real Life Example

A Real Life Example

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?

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

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

Why Talk About This?

- 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

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

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

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)

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)

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 lt 0, N is binomial, if c gt 0, N is negative

binomial

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

Why Does This Matter?

- Under suitable assumptions the Heckman Meyers

algorithm gives the following - E(T) E(N)E(X)
- Var(T) ?(1b)E(X2)?2(bcbc)E2(X)
- Notice if bc0 then
- Var(T) ?E(X2)
- Average, T/N will have a decreasing variance as

E(N)? is large (law of large numbers)

Why Does This Matter?

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

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

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

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

An Example(Continued)

Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method Reserve Estimates by Method

Accident Paid Paid Paid Paid Adjusted CD Adjusted Paid CD Adjusted Paid CD Adjusted Paid CD Adjusted Paid

Year Incurred Devel. Sev. Pure Prem. Hindsight Incurred Devel. Sev. Pure Prem. Hindsight

1986 744 2,143 1,760 1,909 1,687 394 1,936 1,842 1,950 675

1987 2,335 6,847 5,583 5,128 5,128 2,348 6,000 5,790 5,220 2,301

1988 8,371 19,768 16,246 13,451 14,428 10,391 17,352 16,433 13,399 8,001

1989 25,787 44,631 36,887 29,232 32,199 26,048 39,241 36,431 28,512 19,174

1990 60,211 83,760 73,987 61,846 62,974 55,734 79,667 70,246 57,192 43,286

1991 83,093 130,907 95,283 95,185 78,616 79,573 154,268 87,625 84,688 72,157

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

An Example(Continued)

Accident Reserve Reserve Unpaid Single Claim Single Claim Implied

Year Selected Std. Dev. Counts Average Std. Dev. c Value

1986 1,357 637 106 12,802 18,913 0.190

1987 4,260 1,620 330 12,909 19,072 0.135

1988 12,866 3,525 926 13,894 20,527 0.072

1989 30,212 6,428 1,894 15,951 23,566 0.044

1990 62,516 10,198 3,347 18,678 27,595 0.026

1991 90,014 19,166 4,071 22,111 32,666 0.045

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

An Example(Continued)

Accident Severity Severity Estimate

Year Selected Fitted of 1/b

1986 7,723 7,780 0.993

1987 8,501 8,196 1.037

1988 9,577 8,634 1.109

1989 9,919 9,095 1.091

1990 10,739 9,581 1.121

1991 12,194 10,093 1.208

Variance 0.019

An Example(Continued)

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