random, shmandom – just gimme the number maf 3/22/2007 rodney kreps
TRANSCRIPT
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.
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.
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.