Determining Reserve Ranges

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Determining Reserve Ranges

• CLRS 1999
• by
• Rodney Kreps
• Guy Carpenter Instrat

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Why cant you actuaries get the reserves right?
Feel like a target?
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What are Reserves?

• Actual Dollars Paid.
• Distribution of Potential Actual Dollars Paid.
• Locator of the Distribution of Potential Actual
  Dollars Paid.
• An esoteric mystery dependent on the whims of the
  CFO.

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And the Right Answer -

• ALL of the above.

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Actual Dollars Paid

• Only true after runoff.
• Gives a hindsight view.
• Lies behind the question
• Why cant you get it right?

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Distribution of Potential Actual Dollars Paid

• All planning estimates are distributions.
• ALL planning estimates are distributions.
• ALL planning estimates are DISTRIBUTIONS.
• Basically, anything interesting on a
  going-forward basis is a distribution

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Distributions frequently characterized by locator
and spread

• However, the choice of these is basically a
  subjective matter.
• Mathematical convenience of calculation is not
  necessarily a good criterion for choice.
• Neither is Gramps did it this way.

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Measures of spread

• Standard deviation
• Usual confidence interval
• Minimum uncertainty

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Standard deviation

• Simple formula.
• Other spread measures often expressed as plus or
  minus so many standard deviations.
• Familiar from (ab)normal distribution.

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Usual confidence interval

• Sense is, How large an interval do I need to be
  reasonably comfortable that the value is in it?
• E.g., 90 confidence interval. Why 90?
• Why not 95? 99? 99.9?
• Statisticians canonical comfort level seems to
  be 95.
• Choice depends on situation and individual.

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Minimum uncertainty

• AKA Intrinsic uncertainty, Softness, or
  Slop.
• All estimates and most measurements have
  intrinsic uncertainty.
• The stochastic variable is essentially not known
  to within the intrinsic uncertainty.
• Sense is, What is the smallest interval
  containing the value?

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Minimum uncertainty (2)

• How little can I include and not be too
  uncomfortable pretending that the value is inside
  the interval?
• Plausible choice Middle 50.
• Personal choice Middle third.
• Clearly it depends on situation and individual.

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E.g. Catastrophe PML

• David Miller paper at May 1999 CAS meeting.
• Treated only parameter uncertainty from limited
  data.
• 95 confidence interval was factor of 2.
• Minimum uncertainty was 30.

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Locator of the Distribution of Potential Actual
Dollars Paid

• Cant book a distribution.
• Need a locator for the distribution.
• Actuaries have traditionally used the mean.
• WHY THE MEAN?

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WHY THE MEAN?

• It is simple to calculate.
• It is encouraged by the CAS statement of
  principles.
• It is safe - Nobody ever got fired for buying
  IBM.

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Some Possible locators

• Mean
• Mode
• Median
• Fixed percentile
• Other ?!!

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How to choose a relevant locator?

• Example bet on one throw of a die whose sides
  are weighted proportionally to their values.
• Obvious choice is 6.
• This is the mode.
• Why not the mean of 4.333?
• Even rounded to 4?

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What happened there?

• Frame situation by a pain function.
• Take pain as zero when the throw is our chosen
  locator, and 1 when it is not.
• This corresponds to doing a single bet.
• Minimize the pain over the distribution
• Choose as locator as the most probable value.

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Generalization to continuous variables

• Define an appropriate pain function.
• Depends on business meaning of distribution.
• Function of locator and stochastic variable.
• Choose the locator so as to minimize the average
  pain over the distribution.
• Statistical Decision Theory
• Can be generalized many directions
• Parallel to Hamiltonian Principle of Least Work

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Claim All the usual locators can be framed this
way

• Further claim this gives us a way to see the
  relevance of different locators in the given
  business context.

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Example Mean

• Pain function is quadratic in x with minimum at
  the locator
• P(L,X) (X-L)2
• Note that it is equally bad to come in high or
  low, and two dollars off is four times as bad as
  one dollar off.

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Squigglies Proof for Mean

• Integrate the pain function over the
  distribution, and express the result in terms of
  the mean M and variance V of x. This gives Pain
  as a function of the Locator
• P(L) V (M-L)2
• Clearly a minimum at L M

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Why the Mean?

• Is there some reason why this symmetric quadratic
  pain function makes sense in the context of
  reserves?
• Perhaps unfairly ever try to spend a squared
  dollar?

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Example Mode

• Pain function is zero in a small interval around
  the locator, and 1 elsewhere.
• Generates the most likely result.
• Could generalize to any finite interval (and get
  a different result)
• Corresponds to simple bet, no degrees of pain.

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Example Median

• Pain function is the absolute difference of x and
  the locator
• P(L,X) Abs(X-L)
• Equally bad on upside and downside, but linear
  two dollars off is only twice as bad as one
  dollar off.
• Generates the X corresponding to the 50th
  percentile.

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Example Arbitrary Percentile

• Pain function is linear but asymmetric with
  different slope above and below the locator
• P(L,X) (L-X) for XltL and S(X-L) for XgtL
• If Sgt1, then coming in high (above the locator)
  is worse than coming in low.
• Generates the X corresponding to the S/(S1)
  percentile. E.g., S3 gives the 75th percentile.

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An esoteric mystery dependent on the whims of the
CFO

• What shape would we expect for the pain function?
• Assume a CFO who is in it for the long term and
  has no perverse incentives.
• Assume a stable underwriting environment.
• Take the context, for example, of one-year
  reserve runoff.

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Suggestion for pain function

• The decrease in net economic worth of the
  company as a result of the reserve changes.

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Some interested parties who affect the pain
function

• policyholders
• stockholders
• agents
• regulators
• rating agencies
• investment analysts
• lending institutions

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If the Losses come in lower than the stated
reserves

• Analysts perceive company as strongly reserved.
• Not much problem from the IRS.
• Dividends could have been larger.
• Slightly uncompetitive if underwriters talk to
  pricing actuaries and pricing actuaries talk to
  reserving actuaries.

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If the Losses come in higher than the stated
reserves

• If only slightly higher, same as industry.
• Otherwise, increasing problems from the
  regulators.
• Start to trigger IRIS tests.
• Credit rating suffers.
• Analysts perceive company as weak.
• Possible troubles in collecting Reinsurance, etc.
• Renewals problematical.

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Generic Reserving Pain function

• Climbs much more steeply on high side than low.
• Probably has steps as critical values are
  exceeded.
• Probably non-linear on high side.
• Weak dependence on low side

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Generic Reserving Pain function (2)

• Simplest form is linear on the low side and
  quadratic on the high
• P(L,X) S(L-X) for XltL and (X-L)2 for XgtL

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Made-up example

• Company has lognormally distributed reserves,
  with coefficient of variation of 10.
• Mean reserves are 3.5 and S 0.1 (units of
  surplus).
• Then 10 high is as bad as 10 low, 16 high is
  as bad as 25 low, and 25 high is as bad as 63
  low.
• Locator is 5.1 above the mean, at the 71st
  percentile level.

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. . . ESSENTIALS . . .

• All estimates are soft.
• Sometimes shockingly so.
• The uncertainty in the reserves is NOT the
  uncertainty in the reserve estimator.
• The appropriate reserve estimate depends on the
  pain function.
• The mean is unlikely to be the correct estimator.