Random, Shmandom – Just gimme the number MAF 3/22/2007 Rodney Kreps

The (obvious?) essentials Any measure of interest in the real world has uncertainty, and is not meaningful without it. A single number chosen to represent a random variable will depend on the purpose for which the number is to be used.

A Fundamental Truth In order to be meaningful and useful, any measurement or estimate must also have a sense of the size of its uncertainty. And, different uncertainties are physically and psychologically different situations.

Wait a minute, Dr. Physicist What about the charge on an electron? It is a constant, at least according to most scientists. But, any measurement of it involves intrinsic noise. The measurement has uncertainty (currently 8.5 parts in 100,000,000 relative). To the extent that this is small compared to the question we are asking, the measured value is useful.

We know this We use this knowledge automatically, often without doing a conscious calculation. For example, when crossing the street. Or, 5 yards 15 feet 180 inches.

As CEO, are you indifferent?

Corollaries The statement of the estimate frequently implies the size of the uncertainty, correctly or not. When the uncertainty gets too big, the estimate loses all meaning.

Consequences Any measurement or estimate of interest is a random variable. There may or may not be an explicit distribution to represent it. Most importantly, “THE” number does not exist. If you provide one naively you will get hung out to dry.

However… We often have to provide a number. Hopefully we can also provide some sense of the associated uncertainty. Usually there is some intuition. Assuming we have a distribution, what is the “intrinsic softness” of a number representing it? We do not know a number to better than that. My personal choice is the middle third of the distribution. Hurricanes MPLs are 50% or greater.

An Example 100,000 policies each of which has a 10% chance of a $1,000,000 loss. More severe than Personal Auto, but otherwise not too dissimilar. The aggregate distribution for independent losses has a coefficient of variation of 1%. Do you know any insurer with a PA book that stable? Clearly, parameter risk is substantial.

Sources of Uncertainty, the Usual Mathematical Suspects: Limited and erroneous data. Projection uncertainty in the form of –changing physical and legal conditions. –on-level factors not very accurate. –other parameter risk. Model choices.

Risk more Generally Inherent risk, simply due to the insurance process. Internal risks of mistaken planning or models which do not reflect the current reality. External risks from competition, regulatory intervention, court interpretations, etc. Add your own favorites…

Our “Best” Distribution In principle is the Bayesian posterior after all the sources of parameter and other uncertainty are included. Will rely heavily on intuition. May not have an explicit form, but experienced people have a sense for it. Probably is not uniform, in spite of the accountants’ view.

Three interesting numbers Loss for ratemaking. Liabilities for reserving. Required capital for the projected net income. How do we get these from their distributions?

Ratemaking Since we will pay the total, the mean of the severity times the expected frequency is popular. Loss ratio time series also use the mean. Regulators, political appointees, and history look favorably on the mean. The mean it is.

Reserving As a balance sheet item, the estimate should ideally reflect the economic position of the company. SAP do not really allow that, and there are analyst and internal pressures to push reserves from the ideal. But if we at least start from there, what would it be?

Best Reserving Estimate Take an annual context, where each year the prior years’ reserves are restated to their correct value and we are setting current reserves. Assume we have a distribution of outcomes and our job is to pick an estimate that will be as close as possible on restatement. What does this mean?

Least Pain Define a function which describes the pain to the company on the estimate being wrong. Candidate: the decrease in value upon announcement of the restatement. Overestimating reserves is not as bad as underestimating. Pain function is not linear.

What should the pain reflect? The reserve estimator is supposed to display the state of the liabilities for public consumption. The pain should depend upon the deviation of the realized state from the previously estimated state in some quantitative fashion.

Recipe For every fixed estimator, integrate the pain function over the distribution to get the average pain for that estimator. Choose the estimator which gives the minimum pain.

Mathematical representation f(x) – the distribution density function p( ,x) – the relative pain if x ≠  Choose the pain to represent business reality. P(  ) = ∫ p( ,x) f(x) dx – the average of the pain over the distribution Choose  so as to minimize the average pain.

Claims for this Recipe All the usual estimators can be framed this way. This gives us a way to see the relevance of different estimators in the given business context.

Example: Mean Pain function is quadratic in x with minimum at the estimator: p( ,X) = (X-  )^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. Is there some reason why this symmetric quadratic pain function makes sense in the context of reserves?

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 estimator: P(  ) = V + (M-  )^2 Clearly a minimum at  = M

Example: Mode Pain function is zero in a small interval around the estimator, and 1 elsewhere, higher or lower. The estimator is the most likely result. Could generalize to any finite interval. Corresponds to a simple bet with no degrees of pain.

Example: Median Pain function is the absolute difference of x and the estimator: p( ,X) = Abs(  -L) Equally bad on upside and downside, but linear: two dollars off is only twice as bad as one dollar off. The estimator is the 50th percentile of the distribution.

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

Decision Functions for common statistics

Reserving Pain function Climbs much more steeply on the high side than on the low. Probably has steps as critical values are exceeded. Is probably non-linear on the high side. Underestimation is serious. Has weak dependence on the low side. Overestimation is not as serious.

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

And the mean? The pain function for the mean is quadratic and therefore symmetric. It gives too much weight to the low side Consequently, the mean estimate is almost surely too low.

Required Capital Really, this is backwards because usually the capital is fixed and the underwriting and investment are limited by it. The question is “how dangerous is our projected net income distribution?” Again, we can define a pain function to be integrated over the distribution. The pain will depend on the distribution values compared to the available surplus.

Riskiness Leverage A generic form of pain function that can be arbitrarily allocated in an additive fashion. The usual measures for managing to impairment (or insolvency) are all special cases. For actuaries, we frame it in terms of net loss, so that negative values are good. For most people, this does not make sense.

Riskiness Leverage Examples x L TVaR: L x Semi- variance: VaR: L x 

Generic Riskiness Leverage Should be a down side measure (the accountant’s point of view); Should be more or less constant for excess that is small compared to capital (risk of not making plan, but also not a disaster); Should become much larger for excess significantly impacting capital; and Should not increase for excess significantly exceeding capital – once you are buried it doesn’t matter how much dirt is on top. Note: the regulator’s leverage increases.

A miniature company portfolio example using TVAR ABC Mini-DFA.xls is a spreadsheet representation of a company with two lines of business, available online. How do we as company management look at the business? “I want the surplus to be a prudent multiple of the average horrible year.” What is the average horrible year? The worst x%? What is prudent? 1.5, 2, 5, 10?

Conclusions The reality is that numerical answers to interesting questions are always random variables. There is no one number which represents a distribution. There may be a best number for a given purpose. Outcomes always have uncertainty, which can be approximately estimated. This is not the same as the uncertainty in the number representing the distribution.