Non-Gaussian Multivariate Statistical Models and their Alications May 19-24, 2013 Some Challenges in Portfolio Theory and Asset Pricing Chris Adcock - 1
Introduction Finance is a very large subject! This is a ersonal view: Portfolio theory. Asset ricing. Modelling returns. A little otion ricing theory. Many exclusions - certain to be issues omitted which someone here finds imortant! - 2
Structure Background standard stuff. Background skew-ellitical stuff. Skew- ellitical asset ricing models. Challenges Skew-ellitical distributions. Challenges Coulas and Stein s Lemma. Challenges General. - 3
Background Standard Stuff Asset returns have a multivariate normal distribution, X N(,Σ) ~ μ. T T T Portfolio return is an affine transformation, w X ~ N( w μ, w Σ w) X =. Traditionally, investors maximise exected (quadratic) utility of return T ( ) θx 2( X ) 2 w X = 1. U µ This leads to Markowitz efficient frontier and the Caital Asset Pricing Model: ( r ) µ - r = β µ f m f. The conditional distribution of X given X = x is N() and linear in x. - 4
Background Standard Stuff and. This in turn motivates, to some extent, Arbitrage Pricing Theory Now X = β F ε ; wε 0, E wε 0. i j ij i i i i i + = j= 1 i= 1 i= 1 Many quants seek to find better utility functions! But thanks to Stein (1981), Liu (1984), Landsman and Nešlehová (2008): The standard stuff works for all ellitically symmetric distributions and for all utility functions. The efficient frontier, CAPM, APT, linear models. - 5
Background Skew-ellitical stuff Is there a single mean-variance-skewness efficient surface for all exected utility maximisers? Yes, under some skew-ellitical distributions: skew-normal, skew-student. And under distributions of the class introduced in finance by Simaan (1987, 1993). X = U + λv. X has a multivariate ellitically symmetric distribution and V 0 has an unsecified distribution. BTW, it very likely works for all skew-ellitical distributions. Zinoviy Landsman should be working on finer details of the roof! - 6
Skew- ellitical asset ricing models The conditional distributions are non-linear. It is ossible to have exected returns which are better or worse than the CAPM, solely because of skewness. Residual risk in the APT model may be non-zero, which undermines it as a ricing model. These are challenges in finance rather than statistics. - 7
Challenges Skew-ellitical distributions How many hidden truncated variables? Some early evidence suggests more than one work with Martin Eling and Nic Loerfido. Temoral aggregation, leading to more than one truncated variable Challenges in estimation and inference. Distribution theory: how to aroximate sum of truncated normal variables, for examle. Time series effects ARMA-GARCH models. - 8
Challenges Coulas and Stein s Lemma Under normal, Student, or any ellitically symmetric coula, utility function is U n ( X ) U w X, X = h ( Y ), Y ~ ellsym() = = i 1 What haens to Stein s lemma? Is it ossible that some utility functions may be better than others? Same question under skew-ellitical coulas any coula? What is the conditional distribution of AX given BX? i i i i i. - 9
Challenges General Peole build several models, comute some test statistics, then ick the best. Better methods to comare models from different families. For examle, ick model A if it is significantly better than B, C and so on. Otherwise ick the simlest one! Dealing with many variables, for examle 500 in the S&P500. Continuous time models - stochastic PDEs? - 10
And Finally.. In the medium and long term, great innovations can have the effect of stifling future develoments! Are there other aroaches to constructing multivariate models that we should not neglect? From my oint of view: Different marginal distributions. Tractable conditionals. Temoral aggregation & time series effects. Mean-variance-skewness surface(s). - 11
Thank you very much. - 12