Determining Reserve Ranges CLRS 1999 by Rodney Kreps Guy Carpenter Instrat

Why can’t you actuaries get the reserves right? Feel like a target?

What are Reserves? 1Actual Dollars Paid. 2Distribution of Potential Actual Dollars Paid. 3Locator of the Distribution of Potential Actual Dollars Paid. 4An esoteric mystery dependent on the whims of the CFO.

And the Right Answer - ALL of the above.

Actual Dollars Paid Only true after runoff. Gives a hindsight view. Lies behind the question “Why can’t you get it right?”

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

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.”

Measures of spread Standard deviation Usual confidence interval Minimum uncertainty

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

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.

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?”

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.

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%.

Locator of the Distribution of Potential Actual Dollars Paid Can’t book a distribution. Need a locator for the distribution. Actuaries have traditionally used the mean. 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.”

Some Possible locators Mean Mode Median Fixed percentile Other ?!!

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?

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.

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

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.

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.

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

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?

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.

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.

Example: Arbitrary Percentile Pain function is linear but asymmetric with different slope above and below the locator: P(L,X) = (L-X) for X L If S>1, then coming in high (above the locator) is worse than coming in low. Generates the X corresponding to the S/(S+1) percentile. E.g., S=3 gives the 75th percentile.

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.

Suggestion for pain function: The decrease in net economic worth of the company as a result of the reserve changes.

Some interested parties who affect the pain function: policyholders stockholders agents regulators rating agencies investment analysts lending institutions

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.

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.

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

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 X L

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.

... 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.