Multi-Asset Risk Models - CDAR
Portfolio & Risk Analytics ResearchMulti-Asset Risk ModelsOvercoming the Curse of DimensionalityJose MencheroHead of Portfolio Analytics Researchjmenchero@bloomberg.net
Outline Motivation The “curse of dimensionality” Portfolio optimization with noisy covariance matrices Alpha and Hedge portfolios Candidate Multi-Asset Risk Models Ranking the Candidate Models Evaluating the Accuracy of Correlation Forecasts (Factor-pair Portfolios) Evaluating the Quality of Optimized Portfolios (Volatility and Turnover) MAC2 versus MAC1 Comparison Summary2
Portfolio & Risk Analytics ResearchMotivation and Overview
Multi-Asset Factor Covariance Matrices Portfolios may have exposure to multiple asset classes Each asset class is composed of multiple local markets Each local market is explained by many local factorsTo obtain accurate risk forecasts for any portfolio requires acovariance matrix that combines all of the local factorsMAC Covariance MatrixGlobal Equity Block4
The “Curse of Dimensionality” Forecasting accuracy requires a detailed factor structurespanning all markets and asset classes Bloomberg MAC2 covariance matrix contains nearly 2000 factors Portfolio construction demands a robust covariance matrix Risk model should not identify spurious hedges that fail out-of-sample With fewer observations than factors (T K), sample covariancematrix contains one or more “zero eigenvalues” Leads to spurious prediction of “riskless” portfolios This feature makes the sample covariance matrix unsuitablefor portfolio constructionSpecial methods are required to simultaneously provide:1. Accurate volatility forecasts (risk management)2. Robust risk models (portfolio construction)5
Empirical Study 1: The Perils of Noise Take the largest 100 US equities as of 31-Mar-2016, withcomplete daily return history to 13-Jan-1999 Estimate family of asset covariance matrices using EWMA witha variable half-life t Half-life provides convenient “knob” to control noise level Each day, construct the min-vol fully invested portfolio:40 1Bias StatisticVolatility3020222018161014Volatility (%) Bias statistic represents ratio ofrealized risk to forecast risk Biases and out-of-sample volatilitiesincrease dramatically as thecovariance matrix becomes noisier24Bias StatisticΩt 1wt 1 Ωt 11261200255075100125Half Life (days)101506
Turnover and Leverage Minimum-volatility portfolio is fully invested, but not long-only Compute mean turnover and leverage of optimized portfolios1 T N tTOt wt 1 (n) wtt (n)T t 1 n 5100125Half Life (days)Leverage300Daily Turnover (%) Leverage and turnover levelsdiverge at short HL At 10-day HL, leverage is 10x(550% long, 450% short) At 10-day HL, turnoverexceeds 200% per day1 T N tLt wt (n)T t 1 n 17
What Causes this Behavior? Consider two funds: Fund A has volatility of 10%, and expected return of 11.3% Fund B has volatility of 10% and expected return of 8.5% Compute optimal fund weights as a function of the fund correlation As correlation approaches 1, optimal portfolio goes long Fund A andshort Fund B, resulting in large leverage Hedge may not perform well out-of-sample due to estimation error8
Alternative Portfolio-Construction Techniques Alternative portfolio construction techniques: Use optimization but impose constraints (e.g., long only) Forego optimization entirely and use 1/N portfolio (equal weights) For very short HL, the 1/N portfolio indeed outperforms theunconstrained optimal portfolio (but not the long-only portfolio)Observations:24Volatility (%) For well-conditioned covariancematrices, optimized portfolioseasily outperform 1/N portfolio For well-conditioned covariancematrices, unconstrained optimaloutperforms long-only constraint For ill-conditioned matrices,imposing constraints is beneficial26UnconstrainedLong Only2220181/N Portfolio161412100255075100125Half Life (days)1509
Portfolio Optimization (Ex Ante) Decompose optimal portfolio into alpha and hedge portfolios:Ω 1αw α h 1α Ω α Hedge portfolio uncorrelated with theoptimal portfolio:h Ωw 0(Property 1) The hedge portfolio has zero alphah α 0(Property 2) Hedge portfolio is negatively correlatedwith the alpha portfolioh Ωα 0(Property 3)Hedge portfolio reduces portfolio riskwithout changing the expected return10
Geometry of Optimization (Ex Ante) Hedge portfolio is uncorrelated with optimal portfolio P2 2 h2Portfolio Variance Let r h denote the predicted correlation between and h The magnitude of the correlation determines quality of hedge Optimal position in hedge portfolio: h r h11
Potential Pitfalls of Optimization Optimization leads to superior ex ante performance, but is noguarantee of improvement ex post Estimation error within the covariance matrix represents apotential pitfall in portfolio optimization Estimation error in the volatility: Risk models may underestimate the volatility of the hedge portfolio Estimation error in the correlation: Risk models “paint an overly rosy picture” of the correlation betweenthe alpha and hedge portfolios Estimation error gives rise to several detrimental effects: Underestimation of risk of optimized portfolios Higher out-of-sample volatility of optimized portfolios Positive realized correlation between optimized and hedge portfolios12
Estimation Error in the Volatility Suppose we correctly estimate correlation between hedge/alphaportfolios, but we under-estimate volatility of hedge portfolio Side effects: Under-estimation of risk of optimal portfolio Inefficient allocation of risk budget (hedge portfolio adds risk but no return) Increased out-of-sample volatility of optimal portfolio13
Estimation Error in the Correlation Now suppose that we correctly estimatethe volatility of the hedge portfolio However, suppose that we over-estimatemagnitude of correlation Side effects: Under-estimation ofportfolio risk Inefficient allocationof risk budget Increased volatilityout-of-sample Portfolio E is trueoptimal portfolio14
Empirical Study 2 As before, take the largest 100 US equities as of 31-Mar-2016,with complete daily return history to 13-Jan-1999 Estimate a family of asset covariance matrices using EWMAwith a variable half-life t Each day, construct the minimum-volatility portfolio with unitweight in a particular stock:Ωt 1δ nw nt δ n Ωt 1δ nOptimal portfolio (stock n) We have a family of 100 portfolios for each HL parameter Each portfolio is 100 percent long stock a particular stock, andhedges the risk by shorting other stocks15
Biases in Portfolio Volatility Compute mean bias statistics for each of 100 alpha, hedge, andoptimal portfolios Alpha portfolio is fully invested in a single stock Alpha portfolio is independent of the covariance matrix Covariance matrix makes unbiased forecasts for alpha portfolio Hedge/optimal portfolios depend on covariance matrixBiases in correlations must beresponsible for under-forecastingthe volatility of optimal portfolio5Bias Statistics Hedge portfolio risk is largelyunbiased, except for very short HL Optimal portfolio is significantlyunder-forecast for full range of HL4Optimal PortfolioHedge PortfolioAlpha Portfolio32100255075100Half Life (days)12515016
Biases in Correlations Compute mean correlations ex ante and ex post Optimal/hedge correlation: Ex ante, the correlation is exactly zero Ex post, it is positive, indicating inefficient allocation of the risk budget Problem is exacerbated for short HL Alpha/hedge correlation: The hedge always appearsbetter ex ante than it turnsout to be ex post Gap between ex ante andex post grows larger forshort HL parametersCorrelation1.0Optimal/Hedge (ex post)0.5Optimal/Hedge (ex ante)0.0-0.5Alpha/Hedge (ex post)Alpha/Hedge (ex ante)-1.00255075100125150Half Life (days)17
Alpha/Hedge Portfolios versus Efficient Frontier Alpha/hedge decomposition is not the two-fund separation theoremForm the minimum-volatility fully invested portfolio with fixed expected return Universe is largest 100 stocks with covariance matrix using 150-day HL on 31-Mar-2016 Red line is efficient frontier assuming CAPM:E rn n E RM 12E rn n E RM nC is the min-vol fully invested portfolioY/Z are the zero-beta portfolios on theefficient frontier (relative to P/M)P/M are the efficient portfolios usingrespective return assumptions is the alpha portfolio (177% long)H is the hedge portfolio (77% short)Beta (CAPM)Alpha10Blue line is efficient frontier with alphas:Return (%) Efficient FrontierP86M42CY00Note: exceptional returns ( n) are drawnfrom a standard normal of width 30 bps5HZ1015202530Risk (%)18
Constructing the Efficient Frontier Portfolio optimization with equality constraints:Minimize:w ΩwSubject to:Aw b Solution obtained using Lagrange multipliers:w Ω A AΩ A b 1 1 1 Target return and full investment constraints: w1 α w P (Target return) N w2 P 1 2 1 1 1 1 1 w 1 (Full investment) wN Terminology: alphas represent expected stock returns: n n E RM nThe n represent exceptional returns,commonly referred to as “alphas”19
Portfolio & Risk Analytics ResearchCandidate Models
Separating Volatilities and Correlations Divide the task of constructing a factor covariance matrix intotwo parts: Estimating the factor volatilities V (diagonal matrix) Estimating the factor correlation matrix C Factor volatilities are typically estimated using a relativelyshort half-life parameter (i.e., responsive forecasts) Factor correlations typically use longer half-life parameters Reduces noise in the correlation matrix Produces accurate risk forecasts The factor covariance matrix is easily reconstructed:F VCV Present study focuses on comparing the model quality ofcorrelation matrices for the equity block21
Sample Correlation Matrix Sample correlation matrix possesses many attractive properties: Provides arguably the best estimate for any pairwise correlation Best Linear Unbiased Estimate (BLUE) under standard econometric assumptions Gives intuitive and transparent estimates, since it is based on the“textbook” definition of correlation coefficient Produces accurate risk forecasts for most portfolios (with the notableexception of optimized portfolios) Sample correlation matrix also possesses an “Achilles heel”: If there are K factors and T periods, then sample correlation matrixcontains zero eigenvalues (i.e., rank-deficient matrix) whenever T K Rank-deficient matrices predict the existence of “phantom” risklessportfolios that do not exist in reality Sample correlation is not robust for portfolio constructionObjectives: (a) correlation estimates should closely mimic the sample,and (b) provide robust forecasts for portfolio construction purposes22
Other Techniques for Correlations Principal Component Analysis (PCA) Statistical technique to extract global factors from the data Assume a small number of global factors (principal components) fullycapture correlations of local factors (i.e., uncorrelated residuals) Random Matrix Theory (RMT) Statistical technique similar to PCA (factors extracted from data) Eigenvalues beyond a cutoff point are simply averaged Time-series Approach Specify “global” factor returns to explain “local” factor correlations Estimate the exposures by time-series regression Eigen-adjustment Method Eigenvalues of sample correlation matrix are systematically biased Adjust the eigenvalues to remove biasesMenchero, Wang, and Orr. Improving Risk Forecasts for Optimized Portfolios,Financial Analysts Journal, May/June 2012, pp. 40-5023
Blended Correlation Matrices Ledoit and Wolf (2003) showed that blending the samplecovariance matrix with a one-factor model yielded optimizedfully invested portfolios with lower out-of-sample volatility Blend sample correlation (using weight w) with PCA correlationusing J principal components derived from K local factors Specify number of PCA factors by parameter , where J K Two-parameter model for correlation matrix:C B , w w C0 1 w C P Blended Matrix Optimal blending parameters are determined empirically Technique represents the new Bloomberg methodologyLedoit and Wolf. Improved Estimation of the Covariance matrix of Stock Returns,Journal of Empirical Finance, December 2003, pp. 603-62124
Adjusted Correlation Matrices Local models provide our “best” estimates of the correlationmatrices for the diagonal blocks Global model is used to estimate the off-diagonal blocks Diagonal blocks of the global model differ from the correlationmatrices obtained from the local models Integrated model is formed by “adjusting” the global model toreplicate the local models along the diagonal blocks25
Portfolio & Risk Analytics ResearchRanking the CandidateModels
Measuring Biases in Risk Forecasts Bias statistic represents the ratio of forecast risk to realized riskrnt1Biasznt Bn std znt B BnStatistic ntN n If the risk forecasts are exactly correct, the expected value ofthe bias statistic is precisely equal to 1 If the risk forecasts are unbiased but noisy, the expected valueof the bias statistic is slightly greater than 1 Example: suppose we over-forecast volatility by 10% half ofthe time, and under-forecast by 10% half the time221 1 1 1 E B 1.022 0.9 2 1.1 Typical bias statistic for unbiased risk forecasts27
Factor-Pair Portfolios Construct test portfolios capable of resolving minor differencesin volatility forecasts due to differences in correlations Consider factor-pair portfoliosR f1 wf 2 2 12 w2 22 2 r w 1 2 Solve for the weight w that maximizes the percentage of riskdue to the off-diagonal correlation Solution is given by:w 1 2 Portfolio volatility 2 1 1 r 1/228
Description of Study Sample period contains 713 weeks (03-Jan-01 to 27-Aug-14) Model contains K 319 factors spanning nine equity blocks Evaluate accuracy of correlations using factor-pair portfoliosParameters used in Study: Use T 200 weeks (equal weighted) as estimation window For PCA, RMT, and blended matrices Use L 0.25 for local blocks Use G 0.10 for global block For blended correlation matrices Assign 80% weight to the sample (w 0.8) for local blocks Assign 20% weight to the sample (w 0.2) for global blocks Blending parameter selection criteria: Low out-of-sample volatility of optimized portfolios29
Correlation Scatterplots (Diagonal Blocks) Local Blended provides a near “perfect fit” to the sample Eigen method and Local RMT exhibit systematic biasesAnalysis date:27-Aug-2014 Compute B-stats for all factor-pairs with meansample correlation above 0.40 (292 portfolios) Eigen method and Local RMT exhibit biases Sample and Local Blended are near ideal valueModelSampleLocal BlendedEigen methodLocal RMTBias Stats1.0341.0360.9490.946Analysis Period: 03-Jan-2001 to 27-Aug-201430
Correlation Scatterplots (Off-Diagonal) Global Blended provides an excellent fit to the sample Time Series and Global RMT exhibit systematic biasesAnalysis date:27-Aug-2014 Compute B-stats for all factor-pairs with meansample correlation above 0.50 (163 portfolios) Time Series and Global RMT exhibit biases Sample and Global Blended near ideal valueModelBias StatsSample1.034Global Blended1.030Time Series0.958Global RMT0.901Analysis Period: 03-Jan-2001 to 27-Aug-201431
Quality of Optimized Portfolios Construct optimal portfolios for each of K 319 factors andrebalance on a weekly basis (Jan-2001 to Aug-2014) Optimized unit-exposure portfolios have the maximum ex anteinformation ratio (i.e., minimum volatility) 1ΩAA αkwk α k Ω A1α kOptimal portfolio (Model A) Define the mean volatility ratio for Model A1 kAv A RefK k kModel with lowest volatility ratio wins Compute factor turnoverTOt X k ,t 1 X k ,tk1 TO TOtT t32
Ranking the Models (Local Models) Allow hedging using only factors within the same block Convert STD, volatility, and turnover into z-scores Positive z-scores represent “above average” Form composite z-score using weights (0.50, 0.25, 0.25)ModelSampleGlobal PCALocal PCAGlobal RMTLocal RMTTime SeriesEigen-methodLocal BlendedStandard Volatility FactorDeviation ez-scoreVol Ratio Turnover .728-2.0021.559-0.7820.4111.2330.2031.567 Local blended model scored above average on all measures Local blended had the highest composite z-score33
Ranking the Models (Global Models) Allow hedging using all factors within the global block Convert STD, volatility, and turnover into z-scores Positive z-scores represent “above average” Form composite z-score using weights (0.50, 0.25, 0.25)ModelGlobal PCAGlobal RMTTime SeriesGlobal BlendedGlobal PCA (Adj)Global RMT (Adj)Time Series (Adj)Global Blended (Adj)Standard Volatility FactorDeviation Ratio -2.1460.7170.4000.405-0.8690.684z-score z-scorez-scoreVol Ratio Turnover 0.3860.9060.5680.577-0.3470.905-0.4681.055 Global blended (adjusted) had highest composite z-score Unified methodology applied throughout estimation process34
Parameter Selection (US Equities) Leverage/Turnover minimized by few PC and low sample weight STD is minimized by taking many PC and high sample weight Volatility is minimized at intermediate valuesLeverageMu noverw 02.903.954.695.796.758.329.6912.4115.6115.73w 0.203.104.044.725.686.477.648.6310.5914.6015.73w 0.403.634.425.025.836.517.518.4010.4314.5315.73w 0.604.365.045.586.296.927.918.8611.0314.7415.73w 0.805.526.216.777.508.209.3110.3412.4715.1315.73w 5.73Mu (local)0.100.200.300.400.500.600.700.800.901.00w .118w .118w .118w .118w .118Mu atility RatioMu (local)0.100.200.300.400.500.600.700.800.901.00w 8w 00.120.250.380.550.671.011.201.761.370.51w 0.200.120.220.310.430.520.710.811.010.830.51w 0.400.160.220.290.380.440.560.610.700.650.51w 0.600.210.260.300.360.400.460.490.540.560.51w 0.800.280.310.330.360.380.410.430.470.510.51w 1.000.510.510.510.510.510.510.510.510.510.51w .000w .000w .000w .000Standard Deviationw 0w .00035
Benefits of Blending (Local Blocks) Blending sample (w 0.8) with PCA model ( 0.2) produced: More accurate risk forecasts Lower out-of-sample volatility Lower turnover and leverageBias StatisticsVolatility RatioFactor 961.40.40.941.20.920.31.00.900.20.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0Sample WeightSample WeightSample Weight36
Benefits of Blending Form minimum-volatility fully invested portfolio of 100 stocksCompute sample correlation (using 25d and 200d HL)Compute PCA model with one factor (25d and 200d HL)Blend the two covariance matricesOut-of-Sample VolatilityBlending benefit exists for110both models10025-day HLBlended 25d HL performs200-day HL90nearly as well as 200d HL80Model with 25d HL assignsless weight to sample (0.4)70Model with 200d HL assigns60more weight to sample (0.8)0.00.20.40.60.81.0Volatility (bps) Sample Weight37
Portfolio & Risk Analytics ResearchMAC2 versus MAC1Comparison
Bloomberg MAC2 and MAC1 Models MAC1: Computes diagonal blocks using RMT method with shrinkage Computes off-diagonal blocks using the time-series method For equities, “core” factors taken from global equity model e.g., Japan autos is regressed on Japan factor and global auto factor For other blocks, “core” factors are weighted average of local factors e.g., Core factor for oil commodities is weighted average of Brent and WTI “shift” Apply integration matrix to recover the diagonal blocks MAC2: Uses blended methodology for both diagonal and off-diagonal blocks Applies integration matrix to recover local models on diagonal blocks MAC1 and MAC2 models use EWMA with same HL parameters: 26 weeks for volatilities 52 weeks for correlations39
US Equity versus US Fixed Income MAC2 estimates closely mimic the sample correlation MAC1 correlations exhibit considerable biases27-Aug-2014 Results indicate that the time-series method does not fullycapture the correlations across asset classes40
Cross Asset-Class Correlations vs Time Consider the correlation between the US energy factor (equity)and the crude-oil commodity factor (Brent shift) Plot predicted and realized correlations (52w HL)1.0RealizedTime SeriesBlended0.8Correlation Blended approach capturesthe observed relationshipvery closely Time-series methodsystematically underpredicts correlation Suggests missing factors intime-series approach0.60.40.20.02005200720092011Year2013201541
Optimized Factor Portfolios (Off-Diagonal) For each of the 319 local equity factors, compute the volatilityratio between MAC2 and MAC1 for optimized factor portfolios1 kMAC2v MAC1K k k MAC2 Model producedlower volatility in more than80 percent of portfolios The average volatility ratiowas 0.89 Similar results hold forwithin-block optimizationsSample Period: 30-Mar-2005 to 27-Aug-201442
Summary Studied effect of noisy covariance matrices on optimization Introduction of second-generation Bloomberg model (MAC2) Adopted “blank-slate” approach to select the best modelamong a broad set of candidate models New methodology: Two-parameter model uses blended correlations at all estimation levels Parameters are empirically determined Integration matrix is applied to recover local models on diagonal blocks New model closely mimics the sample correlation even acrossdifferent asset classes (e.g., equity versus fixed income) New model guarantees full-rank covariance matrix to providereliable forecasts for portfolio construction43
Portfolio & Risk Analytics ResearchTechnical Appendix
Sample Correlation Matrix Sample period contains 713 weeks (01-03-01 to 8-27-14) Model contains K 319 local factors (for nine equity blocks) Compute sample covariance matrix (F0) over T 200 weeks1Fjk f jt f j f kt f k T 1 t Let S0 be a diagonal matrix of factor volatilities from F0C0 S 0 1F0S 0 1 Sample Correlation MatrixC0 provides an unbiased estimate of pairwise correlationHowever, C0 is rank deficient (119 zero eigenvalues)C0 falsely implies the existence of “riskless portfolios”C0 is not suitable for portfolio optimization purposes45
PCA Correlation Matrices (Global/Local) Transform the sample correlation matrix to diagonal basisColumns of U are eigenvectors of C0D0 U C0 U Keep only the first J components, where J T and J KC UDU U is a KxJ matrix Compute the “idiosyncratic” variance kk 1 diag k C C P UDU ΔCorrelationMatrix Scale PCA correlation matrix with official factor volatilitiesFP SC PSPCA covariance matrix Global PCA refers to PCA technique on all local factors (K 319) Local PCA refers to applying PCA on the diagonal blocks46
Random Matrix Theory (Global/Local) Consider the diagonal matrix D̂: First J elements are largest eigenvalues of sample correlation matrix Remaining K-J elements are the average of remaining eigenvalues Rotate back to the original basisˆ UDUˆ Note: diagonal elements not equal to 1C Scale rows and columns to recover 1 along the diagonalsˆ ˆ 1C R Sˆ 1CSwhere Sˆkk Cˆ kk Scale RMT correlation matrix with official factor volatilitiesFR SC RSRMT covariance matrix Global RMT refers to RMT technique on all local factors (K 319) Local RMT refers to applying RMT on the diagonal blocks47
Time-Series Methods (Full Factor Set) Assume local factors are driven by a small set of global factorsf gB ef is TxK , g is TxJ , B is JxK , e is TxKlocal factorsglobal factorsfactor loadingspurely local For equities, the full set of explanatory variables is given bythe factor returns of a global equity multi-factor model Factor loadings are estimated by time-series regressionB g g g f 1 Define factor covariance matricesg gG T 1e eE T 1D diag E Local factor correlation matrixFFS B GB D C FS S FS1 FFS S FS1Time Series(Full Set)48
Time-Series Methods (Partial Factor Set) Partial-set method mirrors full-set method, except each localfactor is regressed on small subset of global factors For instance, the Japan Automobile factor might only beregressed on two global factors: Japan and Automobiles This results in a sparse factor loadings matrix, B Local factor covariance matrix is given byFPS B GB D The correlation matrix is given byC PS S PS1 FPS S PS1Time Series (Partial Set) Selection of relevant global factors: May contain a significant subjective element Omission of important factors may lead to misestimation of risk49
Eigen-Adjusted Correlation Matrices Menchero, Wang, and Orr (2012) showed that eigenvalues ofsample covariance matrix are systematically biasedD0 U C0 UColumns of U are eigenvectors of C0 Let D0 denote the diagonal matrix of de-biased eigenvalues Perform reverse rotation to original basis:C UD0 U Note: diagonal elements not equal to 1 Scale rows and columns to recover 1 along the diagonalsC E S 1CS 1Eigen-adjusted correlation matrix Eigen-adjusted method is only applicable for the local blocksMenchero, Wang, and Orr. Improving Risk Forecasts for OptimizedPortfolios, Financial Analysts Journal, May/June 2012, pp. 40-5050
Blended Correlation Matrices Ledoit and Wolf (2003) showed that blending the samplecovariance matrix with a factor model yielded optimizedportfolios with lower out-of-sample volatility We blend the sample correlation matrix (using weight w) withthe PCA correlation matrix (using J factors) Specify number of PCA factors by parameter , where J K Two-parameter model for correlation matrix:C B , w w C0 1 w C P Blended Matrix Blending can be applied at either global or local levelLedoit and Wolf. Improved Estimation of the Covariance matrix of StockReturns, Journal of Empirical Finance, December 2003, pp. 603-62151
Adjusted Correlation Matrices Local portfolio managers (e.g., US equities) want the “best”correlation matrix, known as the target correlation matrix Factor correlation matrix may be adjusted so that diagonalblocks agree with the target correlation matrix C11 C12 CT CC22 21Targetcorrelationmatrixˆˆ CC1112ˆC ˆˆ C C2122 Estimatedcorrelationmatrix Define adjustment matrix Aˆ 1/2 C1/2CA 11 11 00 ˆ ACAˆ C A1/2 ˆ 1/2C22 C22 Adjustedmatrix Diagonal blocks now agree with CT, Off-diagonal blocks given by:ˆˆ 1/2Cˆ Cˆ 1/2C1/2C A (2,1) C1/2C22 2221 111152
Alternative portfolio construction techniques: Use optimization but impose constraints (e.g., long only) Forego optimization entirely and use 1/N portfolio (equal weights) For very short HL, the 1/N portfolio indeed outperforms the unconstrained optimal portfolio (but not the long-only portfolio) For well-conditioned covariance
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