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?

underwriting comparison with mean solvency = 1.00

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

95 97 99 101 103 105 107

loss cv = 5%

loss cv = 50%

assets

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< and S*(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 Functionsfor common statistics

mean mode median 75th percentile

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 companyportfolio 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.