Download Probability Weighted Moments

Probability Weighted Moments
Andrew Smith
[email protected]
28 November 2014
Introduction
• If I asked you to summarise a data set, or fit a distribution …
• You’d probably calculate the mean and standard deviation
• … followed by skewness and kurtosis.
• But there is an alternative, popular with hydrologists
• They are called L-moments, or Probability Weighted Moments
• This presentation considers whether L-moments could have a
role in actuarial work.
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Standard Deviation and L-Scale
• Stdev σ = {E(X-µ)2}1/2
• L-scale λ2 = E|X1-X2|/2
• Where µ = E(X)
• For X1, X2 independent
• Or, σ = {E(X1-X2)2/2}1/2
• Or, λ2 = [Emax{X1,X2}-Emin{X1,X2}]/2
• For X1, X2 independent
σ=1
λ2 = 1/√π
Standard normal density
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σ=∞
λ2 = 1/6
Pareto density (α=2)
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Does the choice make any difference?
Expressing λ2 as a multiple of σ
0.3
0.35
0.4
0.45
0.5
0.55
0.6
Generalised
Pareto
0←P2
Uniform 1/√3
Max possible
P3
P4
P6
P10
Exp.
Triang.
Normal 1/√π
EGB2
Laplace
Logistic
Normal 1/√π
Student
0←T2
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T3
T4 T6 T10
4
Sampling Behaviour
• Hydrologists prefer Lmoments because of nice
sampling behaviour
L-moments exist
eg Pareto(2)
– Less sensitive to outliers
– Lower sampling variability
Pearson moments
exist
– Fast convergence to asymptotic
normality
Eg Normal
• Same rationale might apply to
actuarial work.
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Model Mis-Specification Error
• Given sufficient data, we might be able to the L-scale or the standard
deviation reasonably accurately
• But we still face estimation error if we plug that estimate into the wrong
distribution.
• Chebyshev-style inequalities suggest things can go very badly wrong, but
the situation is better if we focus on nice bell-shaped distributions.
• In the next two slides we consider an ambiguity set of models containing
{Weibull, Normal, logistic, Laplace, T3, T4, T6 and T10}.
• We ask how wrong we could be if we try to calculate a 1-in-200 event, with
the right input parameter but the wrong model.
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Impact of Model Uncertainty
Using σ as a Proxy for Value-at-Risk
W-
W+
Return Period (Log Scale)
1000 years
T4
500 years
T3
Watch this box
200 years
Tv refer to Student’s
T distributions with v
degrees of freedom.
W+ and W- are the
right and left tails of a
Weibull distribution
with k = 3.43954
where mean =
median.
100 years
50 years
Consensus region
20 years
10 years
5 years
0
1
2
3
4
5
Number of Standard Deviations
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Impact of Model Uncertainty:
Using λ2 as a proxy for Value-at-Risk
Return Period (Log Scale)
W-
W+
T4
1000 years
T3
500 years
200 years
Watch this box
100 years
50 years
20 years
Consensus region
10 years
5 years
0
2
4
6
8
10
-
Number of L-scales
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How Dispersion Measure affects Model Risk
Ratio of Largest to Smallest by Return Period
300%
These are
the box ratios
in the last
2 slides
Stdev
Lscale
250%
200%
150%
100%
5
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10
20
50
100
200
500
1000
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Measuring Skewness
• Dots show Emin{X1,X2,X3}, Emid{X1,X2,X3}, Emax{X1,X2,X3}
• For standard normal, these are at -3/2√π, 0, 3/2√π
• For Pareto 2, these are at 0.2, 0.6 and 2.2
• Pareto has positive L-skew as 0.2 + 2.2 > 2 * 0.6
Standard normal density
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Pareto density (α=2)
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Comparing Measures of Skewness
Weibull X where Xk ~ Exponential
Weibull Parameter k
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
Mean minus Median
Right skew
Left skew
L-skewness
Pearson skew
Range
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16
14
Kurtosis
Distribution Calibration: Method of Moments
P10
12
10
8
6
Undefined:
P2, P3, P4
T2, T3, T4
Off the scale
P6
Exp
4
-4
-2
Laplace
& T6
2
Logistic
T10
N
0
0
Triang
U
-2
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Skewness
4
12
Distribution Calibration: L-moments
1
L-kurtosis
T2
T3
Laplace
T4
T6
Logistic
T10
Normal
Uniform
0.8
0.6
P2
P3
P4
P6
P10
0.4
0.2
Exp
Triang
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
L-skewness
-0.2
-0.4
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Example Application: Asset Returns
• The simplest, and one of the most widely used asset return models is the geometric
random walk. Returns over disjoint periods are independent (and not necessarily
normally / lognormally distributed)
• Whether we look at returns in absolute or log terms, we can use mathematical
theorems for the Pearson moments or products of random variables, to determine
(for example) moments of one-year-returns from behaviour or one-week returns.
• There is no similar (yet known) theorem for L-moments
• But we could calibrate the weekly distribution using L-moments and then convert to
Pearson moments (using an assumed distribution) to do the risk aggregation.
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Example Application: Collective Risk Theory
• The Cramér-Lundberg (compound Poisson) collective risk model considers
aggregate losses when individual loss amounts are independent observations from
a known distribution and the number of losses follows a Poisson distribution,
independent of the loss amounts
• There are formulas for the Pearson moments of the aggregate loss distribution
given the Poisson frequency and the moments of the individual loss distribution
• There is no similar (yet known) formulas for L-moments
• We could calibrate loss distributions using L-moments, then convert to Pearson
moments using an assumed distribution for the risk aggregation.
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Example Application: ASRF Credit Model
• Vasiček’s Asymptotic Single Risk Factor (ASRF) model is widely used in credit risk
modelling, and also forms the basis of the Basel capital accord for regulatory credit
risk.
• The model is based on a Gauss copula model, where the two inputs are a
probability of default (which turns out to be the mean of the loss distribution) and a
copula correlation parameter ρ, applying to all loan pairs.
• The formula’s derivation uses an expression for the variances of losses conditional
on a single risk factor, which tends to zero for diversified portfolios (Herfindahl index
tends to zero).
• There is no similar (yet known) expression using L-moments.
• As the copula drivers cannot be observed directly, the ρ parameter is conventionally
calibrated by reference to empirical loss distribution properties.
• The standard deviation of the L-scale are equally suitable for this purpose.
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What about Maximum Likelihood?
• In this session, we have compared classical (Pearson)
moments to PWMs.
• More work is required to compare these to alternative methods
such as Maximum Likelihood
• Initial results suggest that
– Max Likelihood has attractive large sample properties if you know the
“true” model
– Practical computational difficulties finding the maxima – the problem
often turns out to be unbounded
– Our methods for examining model mis-specification impact suggest
poor resilience, but this is a function of chosen ambiguity set.
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Comparing Methods - Tractability
Pearson Moments
Probability Weighted Moments
• Analytical formulas known for
many familiar distributions (but
which came first?)
• Unfamiliar, unsupported,
intractable
• Neat proofs known for risk
aggregation calculations
• Taught in statistics courses
• Supported in widely-used
computer software
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• Are these barriers cultural or
technical?
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Comparing Methods: Statistical
Pearson Moments
Probability Weighted Moments
• Consistent, but requires higher
moments to be finite, excluding
members of some distribution
families such as Student T and
Pareto
• Finite mean is sufficient for
estimation consistency
• Sampling error is sensitive to
outliers
• Relatively good at capturing
tails of a distribution
• Less sensitive to outliers
• Uniquely determine a
distribution
• Relatively good at capturing the
middle of a distribution
• Multivariate extensions
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Questions
Comments
Expressions of individual views by members of the Institute and Faculty
of Actuaries and its staff are encouraged.
The views expressed in this presentation are those of the presenters.
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