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?
1.8
underwriting comparison with mean solvency = 1.00
1.6
1.4
loss cv = 5%
1.2
loss cv = 50%
1
assets
0.8
0.6
0.4
0.2
0
95
97
99
101
103
105
107
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(m,x) – the relative pain if x ≠ m
• Choose the pain to represent business
reality.
• P(m) = ∫ p(m,x) f(x) dx – the average of the
pain over the distribution
• Choose m 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(m,X) = (X- m)^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(m) = V + (M- m)^2
• Clearly a minimum at m = 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(m,X) = Abs(m -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(m,X) = (m -X) for X< m and S*(X- m) for X> m
• 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
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
VaR:
L
x
TVaR:
L
x
Semivariance:
L
m
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.