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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

slide10

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

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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

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

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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

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 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

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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