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An Overview of My Research Alex Trindade Associate Prof., Dept. of Mathematics & Statistics, Texas Tech University • Originally from Europe (Portugal). • Grew up in South Africa. • Bachelors in Mathematics, 1988, University of Southampton (England). • Masters in Mathematics, 1992, University of Oklahoma. • Programmer, 1993-1995, IBM (Dallas). • PhD in Statistics, 2000, Colorado State University. • Assistant Prof., 2000-2007, Dept. of Statistics, University of Florida.
Success VaR Failure Risk Measures in Finance X: continuous random variable with pdf f(x) and cdf F(x)=P(X≤x). : a probability level (0< <1). Commonly used in financial risk management: • Value-at-Risk (VaR). • Conditional Value-at-Risk (CVaR), a.k.a. Expected Shortfall CVaR
Why is VaR Important? • In the wake of several spectacular banking collapses, international regulators have decreed that banks must calculate and report the risk of each investment made. • Insurance industry: thinking hard about implementing regulation… • Financial management firms: increasingly interested in assessing their market exposure due to increasing volatility (large fluctuations in exchange and interest rates; exponential growth of derivatives market). • VaR “invented” by the Riskmetrics group at JPMorganChase (a global leader in investment banking), c. 1994. • World leaders in the theory of risk, and risk management: • Europe: The RiskLab at ETH Zurich (Swiss Federal Institute of Technology). Lead by Paul Embrechts. • North America: The Quantitative Analytics group at Standard & Poor’s (NYC). Lead by Craig Friedman. CC20044.03
Why is CVaR Important? Has a number of clear theoretical advantages over VaR. When optimizing portfolios: problem formulations with CVaR constraints far more tractable than VaR-based counterparts. Expect that CVaR-based methods will lead to superior description and mitigation of risk. Tangible difference; would be appreciated by both practitioners and regulators. More research needs to be done; encourage funding agencies like NSF to support such work. CC20044.03
Relationship between VaR and CVaR CC20044.03
Further CVaR Research I’m Thinking About… • Deviation CVaR: Can axiomatize construction of measures of deviation (dispersion); don’t have to be symmetric; generalize variance: • (standard deviation, symmetric) • (deviation CVaR, asymmetric) Can now generalize Markov & Chebyshev Inequalities; Cramer-Rao lower bound; theory of UMVUEs; etc. ? • Deviation CVaR Regression: using deviation CVaR in place of LS gives what kind of estimators? (Ans: same as -quantile regression.) • OptimalDeviation CVaR Prediction:generalize optimal prediction theory based on MSE. • How do properties of estimators of VaR and CVaR compare when we have > such that VaR= CVaR?
Saddlepoint Approximations • Random variable X; moment generating function M(t), K(t)=log M(t). How to get density f(x)? Could use Inversion Formula: • An intractible integral! Using a Laplace approximation to an integral, a saddlepoint approximation to f(x) is a very accurate and (relatively) easy alternative:
Statistical Inference with Saddlepoint Approxs • Parametric inference application: have data (y1,…,yn), parameter of interest, , is solution (root) of a quadratic form in normal random variables • Such Quadratic Estimating Equations (QEEs) arise in many fields in connection with maximum likelihood and method of moments estimators: • Time series models; • Linear and nonlinear regression; • Semiparametric models. • Instead of trying to determine distribution of estimator of , much easier to saddlepoint approximate dist of QEE. (Can then relate dist of QEE to dist of estimator of .) • Construct confidence intervals for by inverting a hypothesis test.
Least Absolute Deviations: (L1 loss.) • Least Squares: (L2 loss.) Quantile Regression Consider observations i=1,…,n from the linear regression model Usually estimate via minimization of some functional of residuals:
10% of points below line: = 0.10 (Give 9 times more weight to negative residuals than to positive residuals.) Quantile Regression - continued • Quantile Regression (Koenker & Bassett, 1978); estimate -quantile surface of Y|x: • Many techniques in OLS still need to be extended for Quantile Regression: • Model selection; • Estimation when close to 0 or 1.
Time Series Analysis Dow Jones Industrial Average: Jan 2008 to present…
Time Series Related Research • Estimation of VaR and CVaR for stationary processes: • best linear prediction with deviation CVaR. • Saddlepoint approximations: • QEEs, etc. • Quantile regression: • connection between quantile AR’s and random coefficient AR’s. • Econometrics: • ARMA’s with asymmetric white noise; ARCH & GARCH models. • Multivariate Time Series: • efficient parameter estimation. • Longitudinal Data: • connections with time series; state-space models.
Something Recent: ARMA with AL noise Bivariate pdf’s of (Xt,Xt-h)
Something Recent: ARMA with AL noise Fitted pdf’s to a real dataset
Who Employs Statisticians? A lot of industries... • Pharmaceutical Companies • Typically based in large metro areas (NE seaboard). • Both M.S. and Ph.D. level. • Median starting salaries as of 2007: • $70,000 (M.A.); $95,000 (Ph.D.). • Large Engineering & Business Companies • Typically based in large metro areas. • Both M.S. and Ph.D. level. • Median starting salaries as of 2007: • $60,000 (M.A.); $80,000 (Ph.D.).
Who Employs Statisticians? • The Federal Government • Primarily based in the Washington D.C. area. • Both M.S. and Ph.D. level. • Median starting salaries as of 2007: • $60,000 (M.A.); $90,000 (Ph.D.). • Colleges & Universities • Widely distributed. • Ph.D. level only. • Median starting salaries as of 2007: • $85,000 (Biostatistics, 12 months) • $72,000 (Statistics, 9 months, University) • $58,000 (Statistics, 9 months, College)
Thank You! • Stop by my office, MATH 211. • firstname.lastname@example.org