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Portfolio Optimization Prof. Daniel P. Palomar ELEC5470/IEDA6100A - Convex Optimization The Hong Kong University of Science and Technology (HKUST) Fall 2020-21 Outline 1 Primer on Financial Data 2 Modeling the Returns 3 Portfolio Basics 4 Heuristic Portfolios 5 Markowitz’s Modern Portfolio Theory (MPT) Mean-variance portfolio (MVP) Global minimum variance portfolio (GMVP) Maximum Sharpe ratio portfolio (MSRP) Outline 1 Primer on Financial Data 2 Modeling the Returns 3 Portfolio Basics 4 Heuristic Portfolios 5 Markowitz’s Modern Portfolio Theory (MPT) Mean-variance portfolio (MVP) Global minimum variance portfolio (GMVP) Maximum Sharpe ratio portfolio (MSRP) Asset log-prices Let pt be the price of an asset at (discrete) time index t. The fundamental model is based on modeling the log-prices yt , log pt as a random walk: yt = µ + yt−1 + ϵt Log−prices of S&P 500 index 2007−01−03 / 2017−12−29 7.8 7.8 7.6 7.6 7.4 7.4 7.2 7.2 7.0 7.0 6.8 6.8 6.6 6.6 Jan 03 Jan 02 Jan 02 Jan 04 Jan 03 Jan 03 Jan 02 Jan 02 Jan 02 Jan 04 Jan 03 Dec 29 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2017 D. Palomar (HKUST) Portfolio Optimization 4 / 74 Asset returns For stocks, returns are used for the modeling since they are “stationary” (as opposed to the previous random walk). Simple return (a.k.a. linear or net return) is pt − pt−1 pt Rt , = − 1. pt−1 pt−1 Log-return (a.k.a. continuously compounded return) is pt rt , yt − yt−1 = log = log (1 + Rt) . pt−1 Observe that the log-return is “stationary”: rt = yt − yt−1 = µ + ϵt 1 Note that rt ≈ Rt when Rt is small (i.e., the changes in pt are small) (Ruppert 2010) . 1D. Ruppert, Statistics and Data Analysis for Financial Engineering. Springer, 2010. D. Palomar (HKUST) Portfolio Optimization 5 / 74 S&P 500 index - Log-returns Log−returns of S&P 500 index 2007−01−04 / 2017−12−29 0.10 0.10 0.05 0.05 0.00 0.00 −0.05 −0.05 Jan 04 Jan 02 Jan 02 Jan 04 Jan 03 Jan 03 Jan 02 Jan 02 Jan 02 Jan 04 Jan 03 Dec 29 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2017 D. Palomar (HKUST) Portfolio Optimization 6 / 74 Non-Gaussianity and asymmetry Histograms of S&P 500 log-returns: Histogram of daily log−returns Histogram of weekly log−returns Histogram of biweekly log−returns 20 25 60 20 15 50 40 15 10 30 Density Density Density 10 20 5 5 10 0 0 0 −0.10 −0.05 0.00 0.05 0.10 −0.2 −0.1 0.0 0.1 −0.3 −0.2 −0.1 0.0 0.1 0.2 return return return Histogram of monthly log−returns Histogram of bimonthly log−returns Histogram of quarterly log−returns 14 10 12 8 8 10 6 6 8 Density Density Density 6 4 4 4 2 2 2 0 0 0 −0.3 −0.2 −0.1 0.0 0.1 0.2 −0.4 −0.2 0.0 0.2 −0.4 −0.2 0.0 0.2 return return return D. Palomar (HKUST) Portfolio Optimization 7 / 74 Heavy-tailness QQ plots of S&P 500 log-returns: D. Palomar (HKUST) Portfolio Optimization 8 / 74 Volatility clustering S&P 500 log-returns: Log−returns of S&P 500 index 2007−01−04 / 2017−12−29 0.10 0.10 0.05 0.05 0.00 0.00 −0.05 −0.05 Jan 04 Jan 02 Jan 02 Jan 04 Jan 03 Jan 03 Jan 02 Jan 02 Jan 02 Jan 04 Jan 03 Dec 29 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2017 D. Palomar (HKUST) Portfolio Optimization 9 / 74 Volatility clustering removed Standardized S&P 500 log-returns: Standardized log−returns of S&P 500 index 2007−01−04 / 2017−12−29 2 2 0 0 −2 −2 −4 −4 Jan 04 Jan 02 Jan 02 Jan 04 Jan 03 Jan 03 Jan 02 Jan 02 Jan 02 Jan 04 Jan 03 Dec 29 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2017 D. Palomar (HKUST) Portfolio Optimization 10 / 74 Frequency of data Low frequency (weekly, monthly): Gaussian distributions seems to fit reality after correcting for volatility clustering (except for the asymmetry), but the nonstationarity is a big issue Medium frequency (daily): definitely heavy tails even after correcting for volatility clustering, as well as asymmetry High frequency (intraday, 30min, 5min, tick-data): below 5min the noise microstructure starts to reveal itself D. Palomar (HKUST) Portfolio Optimization 11 / 74 Outline 1 Primer on Financial Data 2 Modeling the Returns 3 Portfolio Basics 4 Heuristic Portfolios 5 Markowitz’s Modern Portfolio Theory (MPT) Mean-variance portfolio (MVP) Global minimum variance portfolio (GMVP) Maximum Sharpe ratio portfolio (MSRP) Returns of the universe In practice, we don’t just deal with one asset but with a whole universe of N assets. N We denote the log-returns of the N assets at time t with the vector rt ∈ R . The time index t can denote any arbitrary period such as days, weeks, months, 5-min intervals, etc. Ft−1 denotes the previous historical data. Econometrics aims at modeling rt conditional on Ft−1. rt is a multivariate stochastic process with conditional mean and covariance matrix denoted as (Feng and Palomar 2016)2 µ , E [r | F − ] t t t 1 h i , | F − − T | F Σt Cov [rt t−1] = E (rt µt)(rt µt) t−1 . 2Y. Feng and D. P. Palomar, A Signal Processing Perspective on Financial Engineering. Foundations and Trends in Signal Processing, Now Publishers, 2016. D. Palomar (HKUST) Portfolio Optimization 13 / 74 i.i.d. model For simplicity we will assume that rt follows an i.i.d. distribution (which is not very innacurate in general). That is, both the conditional mean and conditional covariance are constant: µt = µ, Σt = Σ. Very simple model, however, it is one of the most fundamental assumptions for many important works, e.g., the Nobel prize-winning Markowitz’s portfolio theory (Markowitz 1952)3. 3H. Markowitz, “Portfolio selection,” J. Financ., vol. 7, no. 1, pp. 77–91, 1952. D. Palomar (HKUST) Portfolio Optimization 14 / 74 Factor models Factor models are special cases of the i.i.d. model with the covariance matrix being decomposed into two parts: low dimensional factors and marginal noise. The factor model is rt = α + Bft + wt, where α denotes a constant vector K ft ∈ R with K ≪ N is a vector of a few factors that are responsible for most of the randomness in the market B ∈ RN×K denotes how the low dimensional factors affect the higher dimensional market assets wt is a white noise residual vector that has only a marginal effect. The factors can be explicit or implicit. Widely used by practitioners (they buy factors at a high premium). Connections with Principal Component Analysis (PCA) (Jolliffe 2002)4. 4I. Jolliffe, Principal Component Analysis. Springer-Verlag, 2002. D. Palomar (HKUST) Portfolio Optimization 15 / 74 Time-series models The previous models are i.i.d., but there are hundreds of other models attempting to capture the time correlation or time structure of the returns, as well as the volatility clustering or heteroskedasticity. To capture the time correlation we have mean models: VAR, VMA, VARMA, VARIMA, VECM, etc. To capture the volatility clustering we have covariance models: ARCH, GARCH, VEC, DVEC, BEKK, CCC, DCC, etc. Standard textbook references (Lütkepohl 2007; Tsay 2005, 2013): H. Lutkepohl. New Introduction to Multiple Time Series Analysis. Springer, 2007. R. S. Tsay. Multivariate Time Series Analysis: With R and Financial Applications. John Wiley & Sons, 2013. Simple introductory reference (Feng and Palomar 2016): Y. Feng and D. P. Palomar. A Signal Processing Perspective on Financial Engi- neering. Foundations and Trends in Signal Processing, Now Publishers, 2016. D. Palomar (HKUST) Portfolio Optimization 16 / 74 Sample estimates Consider the i.i.d. model: rt = µ + wt, N N where µ ∈ R is the mean and wt ∈ R is an i.i.d. process with zero mean and constant covariance matrix Σ. The mean vector µ and covariance matrix Σ have to be estimated in practice based on T observations. The simplest estimators areP the sample estimators: sample mean: µˆ = 1 T r T t=1 t P ˆ 1 T − − T sample covariance matrix: Σ = T−1 t=1(rt µˆ)(rt µˆ) . Note that the factor 1/ (T − 1) is used instead of 1/T to get an unbiased estimator (asymptotically for T → ∞ they coincide). Many more sophisticated estimators exist, namely: shrinkage estimators, Black-Litterman estimators, etc. D. Palomar (HKUST) Portfolio Optimization 17 / 74 Least-Square (LS) estimator Minimize the least-square error in the T observed i.i.d. samples: XT 1 2 minimize ∥rt − µ∥ . µ T 2 t=1 The optimal solution is the sample mean: 1 XT µˆ = r . T t t=1 The sample covariance of the residuals wˆ t = rt − µˆ is the sample covariance matrix: 1 XT Σˆ = (r − µˆ)(r − µˆ)T . T − 1 t t t=1 D. Palomar (HKUST) Portfolio Optimization 18 / 74 Maximum Likelihood Estimator (MLE) Assume rt are i.i.d. and follow a Gaussian distribution: 1 − 1 (r−µ)TΣ−1(r−µ) f(r) = q e 2 . (2π)N |Σ| where µ ∈ RN is a mean vector that gives the location Σ ∈ RN×N is a positive definite covariance matrix that defines the shape.
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  • Optimization of Conditional Value-At-Risk

    Optimization of Conditional Value-At-Risk

    Implemented in Portfolio Safeguard by AORDA.com Optimization of conditional value-at-risk R. Tyrrell Rockafellar Department of Applied Mathematics, University of Washington, 408 L Guggenheim Hall, Box 352420, Seattle, Washington 98195-2420, USA Stanislav Uryasev Department of Industrial and Systems Engineering, University of Florida, PO Box 116595, 303 Weil Hall, Gainesville, Florida 32611-6595, USA A new approach to optimizing or hedging a portfolio of ®nancial instruments to reduce risk is presented and tested on applications. It focuses on minimizing conditional value-at-risk CVaR) rather than minimizing value-at-risk VaR),but portfolios with low CVaR necessarily have low VaR as well. CVaR,also called mean excess loss,mean shortfall,or tail VaR,is in any case considered to be a more consistent measure of risk than VaR. Central to the new approach is a technique for portfolio optimization which calculates VaR and optimizes CVaR simultaneously. This technique is suitable for use by investment companies,brokerage ®rms,mutual funds, and any business that evaluates risk. It can be combined with analytical or scenario- based methods to optimize portfolios with large numbers of instruments,in which case the calculations often come down to linear programming or nonsmooth programming. The methodology can also be applied to the optimization of percentiles in contexts outside of ®nance. 1. INTRODUCTION This paper introduces a new approach to optimizing a portfolio so as to reduce the risk of high losses. Value-at-risk VaR) has a role in the approach, but the emphasis is on conditional value-at-risk CVaR), which is also known as mean excess loss, mean shortfall, or tail VaR.