Probability Weighted Moments Andrew Smith andrewdsmith8@deloitte.co.uk 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. 28 November 2014 2
Standard Deviation and L-Scale Stdevσ= {E(X-µ) 2 } 1/2 Where µ = E(X) Or, σ = {E(X 1 -X 2 ) 2 /2} 1/2 For X 1, X 2 independent L-scale λ 2 = E X 1 -X 2 /2 For X 1, X 2 independent Or, λ 2 = [Emax{X 1,X 2 }-Emin{X 1,X 2 }]/2 σ = 1 λ 2 = 1/ π σ = λ 2 = 1/6 Standard normal density Pareto density (α=2) 28 November 2014 3
Does the choice make any difference? Expressing λ 2 as a multiple of σ 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Generalised Pareto 0 P2 P3 P4 P6 P10 Exp. Triang. Uniform 1/ 3 Max possible EGB2 Normal 1/ π Laplace Logistic Student Normal 1/ π 0 T2 T3 T4 T6 T10 28 November 2014 4
Sampling Behaviour Hydrologists prefer L- moments because of nice sampling behaviour Less sensitive to outliers Lower sampling variability Fast convergence to asymptotic normality L-moments exist eg Pareto(2) Pearson moments exist Eg Normal Same rationale might apply to actuarial work. 28 November 2014 5
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. 28 November 2014 6
Impact of Model Uncertainty Using σ as a Proxy for Value-at-Risk W- W+ Return Period (Log Scale) 1000 years 500 years 200 years 100 years 50 years 20 years 10 years 5 years Watch this box Consensus region T4 T3 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. 0 1 2 3 4 5 Number of Standard Deviations 28 November 2014 7
Impact of Model Uncertainty: Using λ 2 as a proxy for Value-at-Risk W- W+ Return Period (Log Scale) 1000 years 500 years 200 years 100 years 50 years Watch this box T4 T3 20 years 10 years Consensus region 5 years 0 2 4 6 8 10 - Number of L-scales 28 November 2014 8
How Dispersion Measure affects Model Risk Ratio of Largest to Smallest by Return Period 300% 250% 200% Stdev Lscale These are the box ratios in the last 2 slides 150% 100% 5 10 20 50 100 200 500 1000 28 November 2014 9
Measuring Skewness Dots show Emin{X 1,X 2,X 3 }, Emid{X 1,X 2,X 3 }, Emax{X 1,X 2,X 3 } 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 Pareto density (α=2) 28 November 2014 10
Comparing Measures of Skewness Weibull X where X k ~ 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 L-skewness Right skew Left skew Pearson skew Range 28 November 2014 11
Distribution Calibration: Method of Moments 16 14 12 Kurtosis P10 10 8 6 4 Laplace & T6 2 Logistic T10 0 N Triang U -2 28 November 2014 12 Exp Skewness -4-2 0 2 4 Undefined: P2, P3, P4 T2, T3, T4 Off the scale P6
Distribution Calibration: L-moments T2 T3 Laplace T4 T6 Logistic T10 Normal Uniform 1 0.8 0.6 0.4 0.2 0-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-0.2 L-kurtosis P10 P6P4P3 Exp Triang P2 L-skewness -0.4 28 November 2014 13
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. 28 November 2014 14
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. 28 November 2014 16
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?) Probability Weighted Moments Unfamiliar, unsupported, intractable Neat proofs known for risk aggregation calculations Taught in statistics courses Supported in widely-used computer software Are these barriers cultural or technical? 28 November 2014 18
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 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 Multivariate extensions 28 November 2014 19
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. 28 November 2014 20