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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
• Where µ = E(X)
• Or, σ = {E(X1-X2)2/2}1/2
• For X1, X2 independent
• L-scale λ2 = E|X1-X2|/2
• For X1, X2 independent
• Or, λ2 = [Emax{X1,X2}-Emin{X1,X2}]/2
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Standard normal density
σ = 1 λ2 = 1/√π
Pareto density (α=2)
σ = ∞
λ2 = 1/6
Does the choice make any difference?Expressing λ2 as a multiple of σ
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0.3 0.35 0.4 0.45 0.5 0.55 0.6
GeneralisedPareto
EGB2
Student
T3 T4 T6 T10
Normal 1/√π
Laplace Logistic
0←P2 P3 P4 P6 Exp.
Uniform 1/√3Max possible
P10 Triang.
0←T2
Normal 1/√π
Sampling Behaviour
• Hydrologists prefer L-moments because of nice sampling behaviour
– Less sensitive to outliers
– Lower sampling variability
– Fast convergence to asymptotic normality
• Same rationale might apply to actuarial work.
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Pearson momentsexist
Eg Normal
L-moments existeg Pareto(2)
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 UncertaintyUsing σ as a Proxy for Value-at-Risk
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0 1 2 3 4 5
Ret
urn
Per
iod
(L
og
Sca
le)
Number of Standard Deviations
5 years
10 years
20 years
50 years
100 years
200 years
500 years
1000 years
T3
T4
W+W-
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.
Consensus region
Watch this box
Impact of Model Uncertainty:Using λ2 as a proxy for Value-at-Risk
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0 2 4 6 8 10
Ret
urn
Per
iod
(L
og
Sca
le)
Number of L-scales-
5 years
10 years
20 years
50 years
100 years
200 years
500 years
1000 years
T3
T4
W+W-
Consensus region
Watch this box
How Dispersion Measure affects Model RiskRatio of Largest to Smallest by Return Period
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100%
150%
200%
250%
300%
5 10 20 50 100 200 500 1000
StdevLscale
These arethe box ratiosin the last 2 slides
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
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Standard normal density Pareto density (α=2)
Comparing Measures of SkewnessWeibull X where Xk ~ Exponential
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3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
Mean minus Median
L-skewness
Pearson skew
Range
Weibull Parameter k
Right skewLeft skew
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Distribution Calibration: Method of Moments
ExpK
urto
sis
SkewnessU
T10
Laplace & T6Logistic
Triang
Undefined:P2, P3, P4T2, T3, T4Off the scaleP6
-2
0
2
4
6
8
10
12
14
16
-4 -2 0 2 4
P10
N
Distribution Calibration: L-moments
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-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
L-skewness
L-ku
rtos
isExp
Triang
P10P6
P4P3
P2
T2T3
LaplaceT4T6
LogisticT10
NormalUniform
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.
28 November 2014 15
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.
28 November 2014 17
Comparing Methods - Tractability
Pearson Moments
• Analytical formulas known for many familiar distributions (but which came first?)
• Neat proofs known for risk aggregation calculations
• Taught in statistics courses
• Supported in widely-used computer software
Probability Weighted Moments
• Unfamiliar, unsupported, intractable
• Are these barriers cultural or technical?
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Comparing Methods: Statistical
Pearson Moments
• Consistent, but requires higher moments to be finite, excluding members of some distribution families such as Student T and Pareto
• Sampling error is sensitive to outliers
• Relatively good at capturing tails of a distribution
• Multivariate extensions
Probability Weighted Moments
• Finite mean is sufficient for estimation consistency
• Less sensitive to outliers
• Uniquely determine a distribution
• Relatively good at capturing the middle of a distribution
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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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