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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

Milliman USA


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

Milliman USA


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

Milliman USA


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

Milliman USA


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?”

Milliman USA


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

Milliman USA


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

Milliman USA


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

Milliman USA


Conclusions
Conclusions

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

Milliman USA


Conclusions1
Conclusions

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

Milliman USA


Sounds good huh
Sounds Good -- Huh?

  • Relatively straightforward

  • Easy to implement

  • Gets distributions of future payments

  • Job done -- yes?

  • Not quite

  • Why not?

Milliman USA


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?

Milliman USA


Real life example
“Real Life” Example

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

  • Results

Milliman USA




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?

Milliman USA


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

Milliman USA


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

Milliman USA


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

Milliman USA


Discussion
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

Milliman USA


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

Milliman USA


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)

Milliman USA


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)

Milliman USA


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

Milliman USA


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

Milliman USA


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)

Milliman USA


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

Milliman USA


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

Milliman USA


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

Milliman USA


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

Milliman USA


An example continued1
An Example(Continued)

Milliman USA


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

Milliman USA


An example continued3
An Example(Continued)

Milliman USA


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

Milliman USA


An example continued5
An Example(Continued)

Milliman USA


An example continued6
An Example(Continued)

Milliman USA


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


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