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Reserve Uncertainty. by Roger M. Hayne, FCAS, MAAA Milliman USA Casualty Loss Reserve Seminar September 10-11, 2001. 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

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Reserve uncertainty
Reserve Uncertainty

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

Milliman USA

Reserves are uncertain
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
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

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Traditional methods
“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

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Traditional methods1
“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

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Traditional methods2
“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

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Basic assumption
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?”

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Some concepts
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

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Why is this important
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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  • Estimate of mean squared error (mse) of reserve forecast for one accident year:

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  • Estimate of mean squared error (mse) of the total reserve forecast:

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Sounds good huh
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
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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Real life example
“Real Life” Example

  • Paid and Incurred as in handouts (too large for slide)

  • Results

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What happened
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?

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What happened1
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

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What happened2
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
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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  • 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
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

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All may not be lost
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)

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Collective risk model
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)

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Adding parameter uncertainty
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 uncertainty1
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
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 matter1
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
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 example1
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
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 continued1
An Example(Continued)

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An example continued2
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 continued3
An Example(Continued)

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An example continued4
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 continued5
An Example(Continued)

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An example continued6
An Example(Continued)

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Cas to the rescue
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