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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. 28 November 2014 2 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 28 November 2014 σ=∞ λ2 = 1/6 Pareto density (α=2) 3 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 28 November 2014 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. 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 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 28 November 2014 7 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 28 November 2014 8 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 28 November 2014 10 20 50 100 200 500 1000 9 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 28 November 2014 Pareto density (α=2) 10 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 28 November 2014 11 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 28 November 2014 2 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 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 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 28 November 2014 • Are these barriers cultural or technical? 18 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 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